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Cambridge University Press 0521823323 - Automatic Sequences: Theory, Applications, Generalizations Jean-Paul Allouche and Jeffrey Shallit Index More information

Index

abelian cubefree, 37 arithmetic complexity, 340 abelian group, 56 Arno, S., 473 abelian squarefree, 24, 37 Arnold, A., 472, 476 abelian squares, 24, 37 Arnoux, P., 341–343, 473 Aberkane, A., 341 Arnoux-Rauzy sequences, 342 absolutely continuous, 465 array, two-dimensional, 405 accepting state, 128 Arshon word, 206 acceptor, 128 Arshon, C. E., 32, 475 Adams, W. W., 297 Artin, E., 472 Adian, S. I., 34 associate, 66 adjoint, 249 Astoorian, D., 34, 209, 474 Adler, A., 208 asymptotic notation, 63 Agrawal, D. P., 125 Atkinson, K. E., 124 Agronomof, N., 118 Atkinson, M. D., 210 Ahrens, W., 476 atlatl, 471 Akiyama, S., 123, 126, 127 Aubry, S., 468 Alessandri, P., 68, 343 Autebert, J. M., 36 Alexanderson, G. L., 36, 473 automata, cellular, 437 alfalfa, 14 automatic graphs, 171 algebraic, 41, 57 automatic sequences Ali, M. K., 468, 476 closure properties of, 189–197 Allauzen, C., 296 definition of, 152 Allen, Jr., D., 171 subword complexity of, 304 Allouche, J.-P., 27, 32, 34, 36, 121–123, 172, 205, automaticity, 428 207, 209–211, 245, 282, 341, 344, 372, Avgustinovich, S. V., 341, 477 376–378, 404, 426, 427, 437, 454, 467–469, Avishai, Y., 468 471, 473–479 Avizienis, A., 126 almost periodic, 344 avoidance, 24 alphabet, 1 Axel, F., 468, 469 Althöfer, I., 476 Altman, E., 295 B (Barbier infinite word), 114 Alzer, H., 122 Baake, M., 36, 468, 473, 479 ambiguous, inherently, 144 Bach, E., xv ambiguous grammar, 144 Bailey, D., 402 Amou, M., 478 Bak, T. A., xv Anderson, P. G., 297, 476 Baker, A., 67 Angel, E. S., 297 Baker, K. A., 30, 38 antichain, 64 Bakueva, L. G., 469 Apérynumbers, 453 balanced, 313 aperiodic, 32 balanced ternary, 101, 124 Apostolico, A., 31, 35, 125, 246 Balasiński,W., 125 appearance function, 333, 400 Ball, W. W. R., 454 approximable by rationals, 42 Bang, T., 476 Arbib, M. A., 68 Bar-Hillel, Y., 150 Arcelli, C., 297 Barbé,A. M., 426, 427 Ardila, F., 478 Barberye, C., xv

555

© Cambridge University Press www.cambridge.org Cambridge University Press 0521823323 - Automatic Sequences: Theory, Applications, Generalizations Jean-Paul Allouche and Jeffrey Shallit Index More information

556 Index

Barbier infinite word, 114, 299, 339 Blech, I., 467 Barbier, E., 473 Blecksmith, R., 122, 123, 473 Baron, G., 340 Bleuzen-Guernalec, N., 341 Barton, S. K., 123 Blom, G., 471 Baryshnikov, Yu., 343 Bloom, D. M., 67, 342, 472 base for a topology, 5 Blumberg, H., 403 base sequence, 70 Boardman, J. T., 209 base-k representation Boasson, L., 32, 36, 472 canonical, 73 Boffa, M., 208 basic digit set, 101 Böhmer, P. E., 297, 404 Bassily, N. L., 120 Boltzmann constant, 457 Basu, C., 469 Bongiovanni, G., 297 Bateman, P., 115, 473 border, 28, 243 Baum, L. E., 172, 209, 210, 472, 477 Borel, J.-P., 296 Baum–Sweet sequence, 156, 172, 209 Borosh, I., 476 Bazinet, J., 123 Borwein, J. M., 477 Beals, R. M., 376 Borwein, P., 123, 172, 402, 477 Bean, D. A., 30, 37, 478 van den Bosch, P., 126 Beatty, S., 476 Boshernitzan, M., 476 Beauquier, D., 31 Bosma, W., 473 Beauquier, J., 36 bounded gap property, 331 Beck, J., 122 bounded repetition, 36 Becker, P.-G., 377, 454, 478 Bousquet-Mélou,M., 36, 210, 211, 344, 437, 476, 477 Bell numbers, 205 Boute, R. T., 125 Bellissard, J., 468 Bouton, C. L., 454 Bellman, R., 119–121 Bovier, A., 468, 469 Benczur, A., 477 bow ties in Penrose tiling, 455 Bender, E. A., 341 Bowman, D., xv, 117, 297, 374, 473, 476 Bendisch, J., 209 Boyd, D. W., 475 Bendor-Samuel, A., 36 De Bra, P., 472 Berche, B., 469 bracket series, 361 Berche, P.-E., 469 Bradley, D., 115, 473 Bercoff, C., 210 Braggs peaks, 467 Berend, D., 123, 468 Brandenburg, F.-J., 36 Berenstein, C. A., 297 Braunholtz, C. H., 32 Berlekamp, E. R., 454 Breitbart, Y., 437, 478 Berndt, B. C., 472 Bresenham, J. E., 296, 297 Bernoulli, J., 295 Breslauer, D., 31, 35 Bernstein, D., 36 Breusch, R., 67 Berstel, J., xv, 25, 32, 33, 36, 208, 245, 296, 297, 342, Bricard, R., 476 343, 426, 454, 471, 475, 476, 478 Brillhart, J., xvi, 114, 122, 123, 473 Berthé,V., xv, 68, 341–343, 377, 426 Brimkov, V. E., 246 Bertrand-Mathis, A., 123 Brinkhuis, J., 35 Bès, A., 350 de Brito, P. E., 469 Bésineau, J., 120 Britton, J. L., 34 Best, C. A., 121 Brlek, S., 341, 476 β-expansion, 123 Brons, R., 297 Bétréma,J., 209, 475 Brousseau, Br. A., 209 Beukers, F., 453 Brown, Jr., J. L., 125 Bhargava, S., 472 Brown, T. C., 37, 124, 296, 297, 476 bi-infinite word, 31 Brubaker, D. A., 472 bidirectional infinite word, 3 Bruckstein, A. M., 296, 297 bifix-free, 28 de Bruijn, N. G., 343, 467 big-O notation, 63 de Bruijn graph, 339 bijective representation, 101 de Bruijn sequence, 323 binary balanced, 313 de Bruijn word, 323 binding energy, 457 Bruyère,V., 172, 350, 478 Birget, J.-C., 474 Brzozowski, J. A., 31, 35, 36 bisequence, 31 Bucher, W., 36, 150, 475 bispecial subword, 341 Büchi, J. R., 171, 172, 350 Blanc, G., 341 Buck, M., 210 Blanchard, A., 210 Bullett, S., 297 Blanchard, F., 123, 343 Bundschuh, P., 297 Blanchet-Sadri, F., 32 Buneman, P., 209

Index 557

Burd´ık,C., 123 Christol’s theorem, 356 Burnside problem Christol, G., 33, 211, 352, 376, 378 for groups, 34 Chu, D. D., 472 for semigroups, 34 Chu, I.-P., 210 Burnside, W., 34 Chuan, W.-F., 246, 296 Burris, S., 36 Chung, F. R. K., 68 Bush, L. E., 119, 341 Chvátal,V., 476 Byrnes, J. S., 122, 123 Chytil, M. P., 437 cire, 187, 189, 219 Cahn, J. W., 467 Clark, W. E., 473 Cai, T., 121 clearing algorithm, 110 Calkin, N., 478 Clements, G. F., 113, 120, 473 calliope, 5 Clint, M., 209, 476 canonical base-k representation, 73 closed Capocelli, R. M., 125 under the shift, 331 Carlitz zeta function, 360 closed set, 5 Carlitz, L., 122, 123, 125, 360, 473, 478 closure, 7 Carlitz–Goss gamma function, 362, 377 Cobham’s theorem Carlson, F., 404 on k-uniform morphisms, 174 Carpi, A., 33, 35–37, 476 Cobham, A., 150, 172, 209, 211, 245, 246, 282, 341, Cartier operations, 376 350, 404, 476, 477 Cartier operators, 376 coding, 9 Cartwright, M., 67 Cohen, H., 172, 207, 474 Cassaigne, J., 33, 37, 341–344, 426, 437, 468, 477 Cohn, H., 245, 296 Cassels, J. W. S., 68 Coifman, R., 122 Castelli, M. G., 32 Collatz conjecture, 204 Castle, C. M. A., 297 Collatz–Wielandt function, 255 Cateland, E., 123, 124, 172, 211, 426 Collet, P., 209 Cauchy sequence, 63 Colson, J., 124 Cauchy, A., 124 complete DFA, 129 Cayley–Hamilton theorem, 251 complete numeration system, 71 Cayley–Hamilton equations, 465 complete residue system, 102, 104 CD0L sequence, 245 complexity, subword, 298, 341 cellular automata, 427, 437 composite number, 453 Cerný,A.,ˇ 33, 208, 426 Compton, K. J., 475 Cesáromean, 67 computer graphics, 426 CFG, 143 concatenation, 2 CFL, 143 concatenation operator, 1 chain, 49 Condon, A., 437 Chakrabarti, A., 469 conductivity, 465 Chakrabarti, S., 125 conductor, 465 Champernowne, D. G., 478 configuration Chan, T.-H., 209 of a pushdown automaton, 145 Chang, K.-N., 473 of a Turing machine, 147 Chang, R. K., 32 conjugate, 29, 219 characteristic, positive, 58 conjugate transpose, 80 characteristic block, 286 Connell, I. G., 476 characteristic morphism, 285, 294 context-free grammar, 143 characteristic polynomial of a matrix, 250 context-free language, 143, 144 characteristic sequence, 3 context-sensitive grammar, 146 Chebyshev polynomials, 62 context-sensitive language, 146 Chekhova, N., 246, 343 continued fractions Chen, R.-J., 209, 210 finite, 44 Chen, Y.-G., 478 generalized, 65 Cheo, P.-H., 120 infinite, 46 Chérif,H., 377 simple, 45 chess, 26 unusual, 291–293 chevauchement, 33 Conway, J. H., 454 Choffrut, C., 31, 244, 246 Cooper, C. N., 121, 124, 473 Chomsky, N., 150, 151 Copeland, A. H., 478 Chomsky–Schützenberger theorem, 144, 367 Coquet, J., 120, 121, 124, 126 Choquet, G., 118 Cori, R., 37 Christoffel words, 296 Cosnard, M., 471, 478 Christoffel, E. B., 295 Coster, M., 453

558 Index

Coupland, D., 402 DFA, 128 coupling constant, 457 complete, 129 Coven, E. M., 341–343 minimal, 134 Coxeter, H. S. M., 476 DFAO, 138 Crandall, R. E., xv Diamond, H. G., 476 Crisp, D., 296 Didier, G., 343 critical exponent, 24, 27 Dietmeyer, D. L., 125 Crochemore, M., 35, 36, 38, 245, 472 digit set, 70 Crowe, D. W., 209 digital sequence, 83 CSG, 146 Dilcher, K., 476 cube, 14 Dillon, T. S., 209, 210 cubefree, 14, 24, 25, 28, 30, 32, 36 Diophantine approximation, 377 abelian, 37 Dirichlet’s theorem, 48, 280 strongly, 33 distinguishable states, 148 weakly, 33 Diwan, A. A., 476 Cucker, F., 378 Doche, C., 475 Cuevas, E., 469 Domaratzki, M., xv Culik II, K., 35, 36, 244, 245, 476 Dombi, G., 473 Cull, P., 209 Dömösi,P., 32, 150 Cummings, L. J., xv, 35–37, 471, 476 Dorst, L., 297 Currie, J. D., xv, 28, 30, 34–36, 38, 425, 475, 478 double sequence, 405 cutting sequence, 290 doubly infinite word, 31 cyclic chain, 457 Downey, P. J., xvi, 476 cyclic shift, 29 dragon curve, 155 cylinder, 6 drawer, 5 Drazin, M. P., 120 D0L sequence, 245 Dress, F., 209, 476 Damamme, G., 377 Drinfeld modules, 377 Damanik, D., xv, 37, 209, 343, 468, 479 Dringó,L., 124 Danilov, L. V., 297 Drmota, M., 121, 124, 126 darts Droubay, X., 246, 479 in Penrose tiling, 455 Dubey, G. S., 469 Dartyge, C., 31 Dubuc, S., 208 Davies, R. O., 471 Dudeney, H. E., 209 Davio, M., 126 Dudley, U., 121 Davis, C., 210, 474, 475 Dulea, M., 469 Davis, M. D., 125 Dulucq, S., 297 Davison, J. L., 297, 404 Dumas, P., 454 Daykin, D. E., 125 Dumont, J.-M., 124 dead letter, 232 Dupain, Y., 126 Dean, R. A., 36 Durand, F., 350, 477 decision problem, 147 Duverney, D., 477 solvable, 148 Dwork, C., 437, 478 degree of a field extension, 57 Dejean’s conjecture, 30 e Dejean, F., 36 irrationality of, 40 Dekking, F. M., 36, 37, 209, 210, 246, 404, transcendence of, 43, 67 474–476 Ecklund, Jr., E. F., 209 Delange, H., 120, 124 Eckmann, J.-P., 209 Deligne, P., 376, 378, 426 Edlin, A. E., 36 Delyon, F., 468 Edwards, R. E., 121 Demaine, E., 118 Van Eetvelt, P. W. J., 123 Denef, J., 376, 378 Ehrenfeucht, A., 30, 35–37, 246, 341, 342, 472, 477, dense, 49 478 depth, 9 eigenvalue, 250 derivation, 144 Perron–Frobenius, 256, 261 rightmost, 144 eigenvector, 250 derived set, 6 Eilenberg, S., 172, 211, 478 Deschamps, J. P., 126 Ekhad, S. B., 36 Deshouillers, J.-M., 172 Ellis, M. H., 295 deterministic finite automaton, 128 Elqortobi, A., 208 with output, 138 Elser, V., 34–36 Devroye, L., 295 Eminyan,` K. M., 121 Deych, L. I., 469 empty word, 2

Index 559

en passant, 26 Fine–Wilf theorem, 13–14, 26, 32 energy, 457, 464 generalizations of, 32 van Enter, A. C. D., 468, 469 finite automaton, generalized, 131 Entringer, R. C., 36, 37 finite index, 133 entropy, 303 finite-state function, 138 Epifanio, C., 426 finite-state transducer, 140 -free morphism, 31 first difference sequence, 450 -transition, 131 first-digit sequence, 201 equivalence relation, 133 fixed point of a morphism, 10, 23, 173, 175, 212 right-invariant, 133 Flajolet, P., 123, 124, 378 Er, M. C., 209, 210 Fliess, M., 376 Erd˝os,P., 37, 122, 125, 469, 472, 473, 477, 478 Florek, K., 68 Euclidean norm, 80 Flye Sainte-Marie, C., 343 Euclidean ring, 66 Folding lemma, 183 Euler’s constant, 401 Fon-Der-Flaass, D. G., 341, 477 Eulerian cycle, 324 formal Laurent series, 60, 351, 359 Euwe, M., 26, 32, 471 continued fractions for, 61 Evdokimov, A. A., 37 formal power series, 59, 351 Even, S., 403, 477 Cantor, 353 Evey, J., 151 multivariate, 413 expanding morphism, 9 Rudin–Shapiro, 353 exponent, 34 Thue–Morse, 352 exponential of a matrix, 87 transcendence of, 365 extension of field, 57 Formisano, M. R., 37 external field, 457 Forslund, R. R., 125 Foster, D. M. E., 120 F (the so-called Fredholm number), 385, 403 Fournier, J.-C., 209 f (Fibonacci word), 212, 213, 245, 246, 455, 465–468, Fowler, T., 124 479 fractional part, 48 critical exponent of, 246 fractional power, 23 Fabre, S., 350 Fraenkel, A. S., 36, 68, 119, 125, 126, 295, 296, 341, factor, 2 342, 472, 476 European use of, 31 Frame, J. S., 68 factor graphs, 343 Franˇek,F., 343 factorial, 149, 335, 474 Fredholm number, 403 Fagnot, I., 342, 350, 378 Fredholm series, 376 Fan, L., 469 Fredholm, I., 376, 403 Fatou, P., 472 Fredricksen, H., 209, 343 Faucheux, 476 free monoid, 55 Feigenbaum cascade, 209 Freeman, H., 297 Feigenbaum, M. J., 209 Freitag, H. T., 125 De Felice, C., 35 Freivalds, R., 437 Ferenczi, S., 209, 341–343, 404 frequency, 4 Ferns, H. H., 125 logarithmic, 267 fiber, 160, 415 frequency of letters, 247–280 generalized to multidimensional sequences, 408 Fresnel, J., 377 Fibonacci morphism, 212 Friant, J., 474 generalized, 244 Frid, A. E., xv, 341, 343, 344, 473, 477 Fibonacci numbers, 65 Fried, E., 68 Fibonacci representation, 105 Friedman, H. M., 29, 472 Fibonacci word, 339, 455, 465–468 Frobenius, G., 282 almost commutative, 213 Frougny, C., 119, 123, 125 finite, 213 function, finite-state, 138 infinite, 212 Furstenberg, H., 343, 344, 376, 378 subword complexity, 298 Fibonacci-regular sequences, 454 Gaafar, M., 297 field, 56 Gabarró,J., 36, 378 field of fractions, 56 Gabrielian, A., 35, 36 Fife, E. D., 33, 34 Galambos, J., 123 Figà-Talamanca, A., 121 Galil, Z., 31, 35 Filaseta, M., 123 Gambaudo, J.-M., 476 Filipponi, P., 125 γx(w), 331 final state, 128 Gantmacher, F. R., 282 Fine, N. J., 13, 32, 120, 471–473 Gardner, M., 32, 121, 125, 210, 467

560 Index

Garrett, C., 209 Green, J. A., 35 Gasparian, V., 469 Green, R. J., 125 Gaujal, B., 295 Griffith, J. S., 120 Gault, D., 209, 476 Grillenberger, C., 476 Gauss, C. F., 126, 447 Grimm, U., 36, 468 Gaussian integer, 414 Grinberg, V. S., 437 Gazeau, J.-P., 123, 469 Griswold, R. E., 476 Gedeon, T. D., 209, 210 Grossman, E. H., 126 Geelen, J. F., 68 Grounds, W. V., 32 Gelfond, A. O., 120, 122 group, 56 generalized digits, 70 abelian, 56 generalized finite automaton, 131 growing letter, 243 generates, 152 Grundman, H. G., 121 Genolini, C., xv Grundy, P. M., 209 Gerety, C., 209 Grünwald, V., 125 Geshwind, F., 122 Gruska, J., 426 Gessel, I. M., 378 Grytczuk, J., 246, 476 GFA, 131 Guille-Biel, C., 468 Ghez, J.-M., 468, 469 Gumbs, G., 468, 469, 476 Ghosh, A., 469 Gunturk, C. S., 206 Giancarlo, R., 32 Güntzer, U., 473 Gilbert, P., 473 Gupta, N., 34 Gilbert, W. J., xv, 125–127, 172, 211, 426 Gupta, S. K., 123 Ginsburg, S., 150 Guy, R. K., 32, 119, 454 Girgensohn, R., 477 Gyssens, M., 472 Girstmair, K., 477 Glaisher, J. W. L., 115, 473 Hadamard matrix, 424, 451 Glaister, I., 437, 474 Hadamard product, 358 glasses, 455 Hadamard, J., 376 Glaubman, D., 468 Haddad, T. A. S., 469 Goda, M., 469 von Haeseler, F., 205, 426, 427 Godrèche,C., 468 Haferman, J., 407 Golay, M. J. E., 121 Halava, V., 32 Goldstein, S., 124 Hall, Jr., M., 34 Goldstine, J., 36 Halmos, P. R., 31 Golomb, M., 117, 473 halting problem, 148 Golomb, S. W., 473 Halton, J. H., 68 Good, I. J., 32, 343 Hamiltonian, 457 Goodman, T. A., 121 Hamm, D., 246 Goormaghtigh, R., 121 Hamming weight, 119 Goralcik, P., 36, 37 Hamming, R. W., 119 Gossart, C., 126 Hankel determinant, 209 Gottlieb, A. J., 118 Hankel matrix, 208 Gottschalk, W. H., 33, 343, 350 Hanoi, tower of, 177 Goudjil, A., 295 Hansel, G., 172, 350, 478 Gould, H. W., 297 Harase, T., 376, 378, 426 Gouvêa,F. Q., 67, 68 Harborth, H., 471 Gouyou-Beauchamps, D., 297, 378, 477 Hardy, G. H., 67, 68, 119, 404, 477 Grabner, P. J., 120, 123–125 Harju, T., 31, 32, 36, 171, 244, 472 Graham, R. L., 68, 116, 297, 473, 476 Harrison, M. A., 34 grammar Hartmanis, J., 171, 403, 477 ambiguous, 144 Hauschildt, D., 32, 150 context-free, 143 Haussler, D., 36, 150, 475 context-sensitive, 146 Hautus, M. L. J., 378 unambiguous, 144 Hawkins, D., 33 Granville, A., 473 Hayes, P. J., 209 Granville, V., 476 Head, T., 246 Grassl, R. M., 208 Heaslet, M. A., 454 Gratias, D., 467, 469 Heath, F. G., 119 Grazon, A., 475 Heath-Brown, D. R., 69 greatest common divisor, 250 Hecke, E., 404 greedy algorithm, 71 Hedlund, G. A., 32, 33, 35, 295, 340–344, 350, 471, greedy representation, 71 472 Green, A. J. R., 297 Hegstrom, R. A., 32

Index 561

height, 394 indistinguishable states, 148 Heinis, A., 341 induced subshift, 464 Hellegouarch, Y., 377 infinite sequences, 2 Hellerstein, L., 437 infinite words, 2 Hendel, R. J., 246 recurrent, 325 Heppner, E., 120 uniformly recurrent, 328 Herman, G. T., 246 infinity series, 448, 454 Hermisson, J., 468 inflation rules, 208 Hermite, C., 67 Ingalls, C., 172 Hershey, J., 478 inherently ambiguous, 144 Herstein, I. N., 68 initial state, 128 Heuberger, C., 473 insulator, 466 Hickerson, D. R., 473 integers, p-adic, 63 Higgins, P. M., 246 interior sequence, 177 Hilbert’s tenth problem, 148, 451 inverse homomorphism, 9 Hillman, A. P., 473 Iochum, B., 468 Hinz, A. M., 34, 209, 210, 474 irrationality Hirschhorn, M. D., 478 of π,40 Hodgson, B., 350 of e,40 Hof, A., 468 irreducible element, 66 Hoggatt, Jr., V. E., 125, 297 irreducible matrix, 250 Hoit, A., 120 Ising chain, 457 Hollander, M., 119 configuration of, 457 Holton, C., 246, 342, 343 cyclic, 457 Holzer, M., 474 homogeneous, 458 Holzman, R., 68 sites of, 457 homogeneous, 458 transfer matrix of, 457 homomorphism, 8 Ising model, 457 depth of, 9 Ising, E., 467 identity, 23 Istrail, S., 32, 472 inverse, 9 iterated morphisms iterated, 9 music and, 245 length-preserving, 31 Ito, M., 32, 150 nonerasing, 9 Ito, S., 123, 127, 296 overlap-free, 17 Ivanov, S. V., 34 size of, 9 width of, 9 Jackson, D. E., 36, 37 Honkala, J., 171 Jacobs, K., 25, 342, 471 Hopcroft, J. E., 31, 150 Jacobson, N., 68 Horadam, A. F., 476 Janner, A., 468 Hordijk, A., 295 Janssen, T., 468 Hörnquist, M., 469 Jedwab, J., 473 Horváth,G., 32, 150 Jeffreys, H., 67 Horváth,S., 32 Jeˇzek,J., 36 Houselander, L. S., 125 Jiang, S. S., 469 Hu, A., 469 Jiao, Z. K., 469 Huang, D., 469 Jin, G. J., 469 Huang, K., 467 Johansson, M., 469 Huang, Y., 476 Johnson, T., 245 Hubert, P., 246, 342, 343 Johnsonbaugh, R., 210 Huffman, D. A., 150 Joseph, D., 468 hull, 464 Joshi, V. S., 121 Hungerford, T. W., 68 Journal of Integer Sequences, xiii Hurwitz, A., 68 Jungen, R., 376 Jürgens, H., 209 Ilie, L., 32 Justin, J., 32, 37, 343, 477 Il’in, V. I., 469 Iliopoulos, C. S., 246 k-additive function, 120 incidence matrix, 248, 266 k-aligned subword, 4 indegree, 323 k-automatic sequences, definition of, 152 index k-automatic set, 168 infinite, 322 k-block compression, 5 of a finite subword, 322 k-context-free sequences, 208, 340 index of imprimitivity, 256 k-fiber, 160

562 Index

k-kernel, 185, 211 Klop, J. W., 472 k-multiplicative function, 120 Klosinski, L. F., 36, 473 k-quasiautomatic sequence, 435 Kmoˇsek,M., 210 k-recognizable, 172 Knight, M. J., 403 k-regular power series, 446 Knill, O., 468 k-regular sequences, 438 Knopp function, 208 transcendence and, 454 Knuth, D. E., 68, 115, 116, 119, 125, 126, 172, 210, k-uniform morphism, 173 245, 297, 404, 473–476 Kadanoff, L. P., 467 Kobayashi, Y., 33, 34 Kahane, J.-P., 122 Koblitz, N., 67, 68, 473 Kakutani, S., 474 Kobuchi, Y., 246 Kamae, T., 33, 118, 120, 209, 341, 376, 468 von Koch, H., 202 Kanal, L. N., 297 Kochman, F., 341 Kanani, D. V., 125 Kohmoto, M., 467 Kaneps, J., 437 Koksma, J. F., 68 Kano, H., 473 Kolakoski word, 337, 476 Kaprekar, D. R., 121 Kolakoski, W., 476 Karaman, A., 343 Koláˇr,M., 468, 469 Karevski, D., 469 Kolden, K., 68 Karhumäki,J., 31, 33–35, 245, 246, 342, 476, 477 Kolpakov, R., 35 Kari, L., 32 Komatsu, T., 294, 296, 476 Karmakar, S. N., 469 König infinity lemma, 28 Karner, G., 68 Koplowitz, J., 297 Karp, R. M., 437 Korec, I., 427 Karrass, A., 34 Körmendi, S., 126 Kása, Z., 341 Körner, T. W., 121, 122 Kászonyi, L., 32, 150 Korobov, A. N., 478 Kátai, I., 120, 124, 126 Korshunov, A. D., 437 Katok, A. B., 343 Kósa, M., xv Katsura, M., 32, 150 Kosaraju, S. R., 35 Kautz, W. H., 125 Koskas, M., 210, 342, 377, 426 Kawai, H., 120 Kovács, B., 126 Keane, M. S., 208, 342, 343, 476, 477 Krawetz, B., xv Keir, J., xv Krejcar, R., 123 Keller, T. J., 125 Krishnamurthy, E. V., 125 Kelley, J. L., 31 Kristiansen, M., xv Kelly, K., 124 Kritikos, N., 68 Kempner, A. J., 125, 403 Kronecker’s theorem, 49 Kennedy, R. E., 121, 124, 473 Kucherov, G., 35 Keränen, V., 35–37, 472 Kudlek, M., 32, 150 kernel, generalized to two dimensions, 409 Kuich, W., 68, 150 Kervaire, M., 123 Kuipers, L., 68 Ketkar, P., 477 Kullberg, E., 454 Kfoury, A.-J., 33, 36, 471 Kurka, P., 343 Khan, A., 469 Kieffer, J. C., 295 L2 norm, 80 Kim, C. E., 297 of a function, 78 Kim, J. B., 297 Laakso, T., 245 Kimberling’s sequence, 448 Lacampagne, C., 119 Kimberling, C., 125, 448, 454, 476 Lalanne, L., 124 Kirschenhofer, P., 123, 124 Lallement, G., 35 Kitaev, S., 475 Lambek, J., 476 kites, in Penrose tiling, 455 Lambert, J. H., 67 (k, l)-morphism, 187 Landau, E., 341 (k, l)-numeration system, 103, 164 Lando, B., 246 Klarner, D. A., 378, 478 Lanford III, O., 476 Klavˇzar, S., 210 Lang, S., 68 Kleene closure, 7 Lange, E., 427 Kleene’s theorem, 132 Langevin, M., 68 Kleene, S. C., 31, 150 Langford string, 25 Klein, B. G., 246, 282 Langford, C. D., 471 Klein, C. S., 209 language, 2 Kleman, M., 468 context-free, 143, 144 Klepinin, A. V., 37 context-sensitive, 146

Index 563

factorial, 149, 335, 474 linear combination, 56 regular, 8 linearly dependent, 56 languages, 7 linearly independent, 56 product of, 5, 7 Linna, M., 171 quotient of, 8 van Lint, J. H., 68 Larcher, G., 121 Liouville function, 208 Larson, L. C., 36, 473 Liouville number, 43 Lasjaunias, A., 210 Liouville, J., 67 Laubie, F., 296, 372 Lipshitz, L., 376, 378 Laud, P. W., 122, 123 Lisyansky, A. A., 469 Laurent series, 359 little-o notation, 63 continued fractions for, 61 Littlewood, J. E., 121, 122, 404 formal, 60 Liu, N.-H., 469 Laurier, E.,´ 296 Liu, Y., 469 Lavallée,I., 209 Liviotti, E., 469 Lavine, D., 297 locally catenative sequence, 237 Lazarkiewicz, A., 125 Loftus, J., 472 leading-zeros problem, 70 logarithmic frequency, 267, 268, 274, 276, 282 Lecomte, P. B. A., 172 Lomont, J. S., 122, 473 Leconte, M., 35 López, J., 210 Lee, K. P., 342 Lopez, L.-M., 343 Lee, M. H., 469 Loraud, N., 119 Leech, J., 33 Lorentz, R. J., 35 L (left part of an infinite word), 3 Lothaire, M., 31, 32, 123 Legendre, A.-M., 74, 119 Loxton, J. H., 403, 404 Lehmer, D. H., 172, 474 Lu, X.-M., 209, 210 Lehr, S., 403 Lubiw, A., 150, 210 Leighton, W., 210 de Luca, A., 35, 36, 246, 296, 297, 340, 341, Leinfellner, H., 124 476, 477 Leiss, E. L., 209 Lucas, E., 209 Lekkerkerker, C. G., 125 Luccio, F., 297 Lenard, A., 115, 473 Luck, J.-M., 468 length function, 74 Lunnon, W. F., 210, 295, 296, 467, 475 length-preserving morphism, 31 Ly, O., 171 Lengyel, T., 453 Lyndon, R. C., 10, 32, 36, 472 Lennholm, E., 469 Lyndon–Schützenberger theorems, 10–13, 32 Lenz, D., 37, 343 Lysënok, I. G., 34 Lepistö,A., 245, 476 Lepp, B., 116, 473 Ma, H.-R., 469 letters, 1 MacCluer, C. R., 282 LeVeque, W. J., 67 MacMurray, D., 471 Levi’s lemma, 10 Madan, S., 123 Levi, F. W., 32 Maes, A., 350 Levine, D., 467 Magnus, W., 34 Levine, E., 473 Mahler functional equations, 454 Levitt, J., 296 Mahler, K., 32, 67, 68, 118, 377, 403, 404, 477, 478 Levy, L., 209 Mahmoud, B. S., 469 lexicographic order, 5 Main, M. G., 35, 36, 150, 474, 475 Li, S.-Y. R., 208 Majumdar, A. A. K., 209, 210 Li, Z., 121 Makowski, A., 121 Liang, F. M., 68 Mallol, C., 210 Liang, J. J., 473 Mandelbrot, B. B., 426 Liardet, P., 121, 122 Marcus, S., 471, 474 van de Liefvoort, A., 210 Markoff, A. A., 295, 296 limit of a sequence of words, 4 Martin, G., 478 limit point, 6 Martin, J. C., 208 Lin, Z., 468, 469 Martin, M. H., 343 Lindemann, F., 67 Martinez, A. A., xv Lindenbaum, M., 297 Martinez, P., 478 Lindenmayer system, 245 Massarotti, A., 297 Lindenmayer, A., 245, 246 Masser, D. W., 404 Lindhurst, S., xv Massias, J.-P., 341 Lindquist, B., 469 de Mathan, B., 377, 472 Lindström,B., 113, 120, 473 Matiyasevich, Yu. V., 151

564 Index

matrix Moitra, R. K., 469 adjoint, 249 monoid, 2, 55 incidence, 248 free, 55 irreducible, 250 Monteferrante, S. A., 246 non-negative, 250 Montgomery, H. L., 31 permutation, 250 Moore, C., 372 positive, 250 Moore, D., 246, 476 primitive, 31, 250 Moore, E. F., 150 reducible, 250 Moore, E. H., 454 stochastic, 250 Moore, F. R., 474 matrix exponential, 87 moose antlers, 8 matrix norm, 80 Mootha, V. K., 474 Matula, D. W., 125 Morain, F., 473 Mauclaire, J.-L., 120, 121, 124 Moran, W., 296 Mauduit, C., 31, 120, 209, 341, 342, 404 Morcrette, M., 426 Mays, M., 37 Mordell, L. J., 477 McCallum, M., 473 De Morgan, A., 124 McCulloch, W. S., 150 moribund letter, 232 McDaniel, W. L., 121 morphic sequence, 10, 212 McIlroy, M. D., 120, 121, 297 morphism, 8, 31 McMullen, J. R., 121 characteristic, 285, 294 McNaughton, R., 35, 171, 350 depth of, 9 McNulty, G. F., 30, 37, 38, 478 -free, 31 Mealy, G. H., 150 expanding, 9 mean value theorem, 42 identity, 23 measure, 7 iterated, 9 Meijer, H. G., 477 k-uniform, 173 Mellin transform, 124 length-preserving, 31 Mendès France, M., xv, 33, 120–123, 172, 210, 282, nonerasing, 9 363, 376, 377, 403, 467, 468, 472, 475, 479 overlap-free, 17 Mephisto waltz, 25 primitive, 9 Merekin, Yu. V., 476 prolongable, 10, 173, 212 mesh, 49 propagating, 31 Messaoudi, A., 246 size of, 9 metric, 6 squarefree, 35 metric space, 6 Sturmian, 296 Meyer, Y., 122 trivial, 9 Michaux, C., 172, 350, 478 uniform, 9 Michel, P., 282, 342 width of, 9 Microserfs,402 Morris, J., 245, 476 Mi¸ekisz, J., 468, 469 Morrison, D. R., 297 Mientka, W. E., 33 Morse, M., 26, 32, 35, 208, 295, 340–344, 471, 472 Mignosi, F., 32, 37, 246, 296, 297, 342, 343, 426 mortal letter, 9, 212 Miller, W., 342 mortality exponent, 9, 212 Mills, W., 210 Morton, P., 114, 122, 123, 404, 473, 478 Milutinović,U., 210 Moser, L., 476 Minc, H., 31, 282, 476 Mossé,B., 34, 341 minimal, 331 Mossinghoff, M., 123 minimal DFA, 134 Moulin-Ollagnier, J., 37 minimal interior sequence, 177 Mouline, J., 341 minimal polynomial of a matrix, 250 Mourant, W. J., 123, 404, 473, 478 minimal square, 28, 31 move in tower of Hanoi, 177 Minsker, S., 209 legal, 177 Minsky, M., 171, 172, 282 Mozes, S., 426 Mione, L., 341 Mrówka, S., 125 mirror invariance, 29, 339 Mukherjee, M., 68 Mirsky, L., 119 Mullhaupt, A. P., 208, 478 Mitchell, C. J., 473 multigrade, 153 Mitrana, V., 32 multiplicatively dependent, 52 Mkaouar, M., 210 multiplicatively independent, 52 Möbius function, 13 Murata, L., 120, 124 mod as a binary operation, 15, 48 murmur, 471 modular trellis, 426 Murugesan, S., 125 Mogyoródi,J., 120 Mushkin, M., 295, 296, 342

Index 565

music, 33, 245, 448 d’Ocagne, M., 473 Musikhin, S. F., 469 occurrence function, 278 Myhill, J., 150 Odlyzko, A. M., 125 Myhill–Nerode theorem, 133, 137 Oettinger, A. G., 151 generalization of, 149 Oh, G. Y., 469 Okada, T., 124 Nadler, M., 126 Ol’shanskii, A. Yu., 34 Nait Abdallah, A., 426 de Oliveira, C. R., 469 Narbel, P., 343 Olivier, M., 31, 117, 473 Nathanson, M., 341 Olivos, J., 473 Nazareno, H. N., 469 Olson, E. C., 297 Nelson, A. H., 125 Oltikar, S., 121 Nelson, E., 36 ,63 Nerode, A., 150 ω-word, 31 Newman, D. J., 120, 123, 124 of DOL system, 245 Newman, M., 477 open set, 5 NFA, 130 operator norm, 251 NFA-, 131 orbit of an infinite word, 327 Nicol, C., 123 order-preserving, 72 Nicolas, J.-L., 341 Ore’s theorem, 355 Niederreiter, H., 68, 201, 474 Ortuño,M., 469 Nielsen, P. T., 471 Osbaldestin, A. H., 124 nim, 454 Ostlund, S., 467 game of, 450 Ostrowski, A., 126 nim product, 453 Ouchterlony, T., 469 nim sum, 448, 451, 453 outdegree, 323 Nishioka, K., 296, 403, 404, 454 overbar, 16 Nivat, M., 31, 36 overlap, 14 Niven number, 121 overlap-free, 14, 16, 35 Niven, I., 67, 68 Nogueira, A., 342 p (characteristic sequence of the primes), 3, 280 noiselets, 122 p-adic valuation, 39 Noland, H., 32, 209 p-adic integers, 63 nonerasing, 9 p-adic numbers, 62 nonerasing morphism, 9 automatic, 402 non-negative matrix, 250 palindrome complexity, 466 non-negative vector, 250 palindromes, 5, 28 nonrepetitive words, 32 detecting efficiently, 31 strongly, 37 Pan, F. M., 469 nondeterminism, 130 Pandit, R., 467 nondeterministic finite automaton, 130 Pansiot, J.-J., 33, 37, 171, 246, 342, 476 nonzero pattern sequence, 78 paperfolding, 122 Noonan, J., 35 paperfolding sequences, 181–185 Nørg˚ard,P., xv, 33, 448 continued fractions and, 183–185 Nori, F., 468 regular, 155 norm, 107 Papert, S., 171, 172, 282 Euclidean, 80 Parent, D. P., 472 matrix, 80 Parry, W., 123, 208 operator, 251 partial order, 64 submultiplicative, 251 partition function, 457, 458 normal number, 401 Parvaix, B., 342 Novikov, P. S., 34 de Parville, H., 209 number-of-divisors function, 401 Pascal triangle, modulo d, 420 numeration system, 70 Patashnik, O., 297 complete, 71 Patel, R. B., 121 perfect, 71 pattern avoidance, 24 representation in, 70 pattern sequence, 78 unambiguous, 71 nonzero, 78 Nürnberg, R., 208 zero, 78 pattern transform, 82, 119, 449 O() notation, 63 Paul, M., 473 o() notation, 63 Paul, M. E., 343 O’Connor, L., 473 P˘aun, G., 32, 471, 476 O’Keefe, K. H., 125 Pawlak, Z., 125

566 Index

PDA, 145 Pour-El, M. B., 403 Pedoe, D., 454 power series, 351 Peitgen, H.-O., 172, 209, 211, 426, 427 Cantor, 353 Penney, W., 125, 126 multivariate, 413 Pennington, J. V., 245 Rudin–Shapiro, 353 Penrose, R., 467 Thue–Morse, 352 Penrose tiling, 455 Pratt, V., 245, 476 perfect numeration system, 71 prefix, 2 perfect set, 6, 24, 30, 34 proper, 2 perfect shuffle, 5 prefix language, 204 period, 4, 220 Preparata, F. P., 35 least, 4 preperiod, 4 period-doubling sequence, 176, 209, 467 least, 4 periodic, 24 Price, J. F., 121 purely, 4 Priday, C. J., 471 ultimately, 4 primary, 28 periodicity of two-sided infinite words, 220 prime number theorem, 280 Perles, M., 150 PRIMES2 (primes in base 2), 2, 147 permutation matrix, 250 primitive, 344 permutation number, 340 primitive matrix, 31, 250 Perrin, D., 31, 150, 350 primitive morphic, 401 Perron, O., 67, 282 primitive morphism, 9 Perron–Frobenius eigenvalue, 256, 258, 261, 266 primitive polynomial, 374 Perron–Frobenius theorem, 256, 260, 282 primitive words, 13, 29, 32 persistence, 118 enumeration of, 13, 367 Petersen, A., 426, 427 unsolved problem on, 366 Petersen, H., 32, 150, 378 Prodinger, H., 36, 123, 124, 210, 473 Pethö,A., 126 productions, 143 Pettorossi, A., 209 prolongable, 10, 173, 212 Peyrière,J., 209, 246, 282, 426, 468 propagating morphism, 31 Phillips, G. M., 125 proper prefix, 2 π Propp, J., 478 decimal expansion of, 26 Prouhet, E., 32, 208 digits of, 402 pumping lemma, 136, 150 irrationality of, 40 for context-free languages, 148 transcendence of, 43, 67 pure morphic sequence, 10, 212 whether automatic, 402 pure point, 465 Pin, J.-E., 31 purely periodic, 4 Pinho, S. T. R., 469 pushdown automaton, 145 Pirillo, G., 37, 209, 246, 343, 477, 479 Pitteway, M. L. V., 297 Qin, M.-G., 469 Pitts, W., 150 quasiautomatic sequence, 435 Plandowski, W., 35 quasicrystals, 455, 467 Pleasants, P. A. B., 34, 37, 295, 296, 467, 468, 475 in botany, 469 Plouffe, S., xiii, xv, 400, 402 Queffélec,M., 123, 342, 404 Pocchiola, M., 296, 297 quotient, 8 Point, F., 208, 350 Pollak, H. O., 116, 473 r (Rudin–Shapiro sequence), 78, 186 Pollington, A., 296 Rabin, M. O., 35, 150 polynomial, 58 Rabinowitz, S., 473 Pomerance, C., 437 Rabizo, O. V., 469 Poole, D. G., 209 Raimi, R. A., 282 Poon, A., xv Ramshaw, L., 126 van der Poorten, A. J., 68, 210, 296, 403, 404, Rand, D., 467 472, 475 Randall, J., 34, 209, 474 Popken, J., 477 Randé,B., 282, 350, 454 Porges, A., 120 rank of an equation, 42 Porta, H. A., 295, 342, 476 Rao, A. N., 121 positive characteristic, 58 Rao, G. S., 125 positive closure, 8 Rao, M. N., 125 positive matrix, 250 Rasson, J. P., 476 positive vector, 250 rational function, 59 Post, E., 172 Rauzy graphs, 343 Pottle, S., 437 Rauzy, G., 33, 342–344, 376, 477

Index 567

van Ravenstein, T., 68 ring, 56 ray, 340 Euclidean, 66 Ray, G. C., 209, 210 Risley, R. N., 342 Rayleigh, J. W. S., 476 Ritchie, R. W., 171, 172 Razafy Andriamampianina, D., 210 Rivier, N., 469 reachable state, 129 Robbins, D., 210 Read, R. C., 35 Robbins, N., 478 real number Robert, A., 127 algebraic, 41 Roberts, J. A. G., 468 transcendental, 41 Roberts, J. B., 113, 172 Recamán,B., 121 Robson, G. C. A., 209 Recher, F., 377 Robson, J. M., 150, 437 Recht, L., 245 robust, 157 rectilinearly connected, 425 robust letter, 232 recurrence function, 328 robustness of a computational model, 130 recurrent, uniformly, 328 Rockett, A. M., 68 recurrent infinite word, 325 Rohl, J. S., 209 recurrent subword, 324 Rokicki, T. G., 172 recursive, 147 Ronse, C., 297 recursively enumerable, 147 Rosaz, L., 38 reduced adjoint, 250 Rosenfeld, A., 297 reducible matrix, 250 Rosenman, M., 245 Rees, D., 35, 477 Ross, R., 36 reflexive, 133 Rosser, J. B., 69 de Regt, M. P., 125 Rossmanith, P., 474 regular expressions, 8 Rote, G., 342, 477 regular language, 8 Roth, K. F., 67 regular paperfolding sequence, 155, 182, 203, Roth, P., 36, 37, 476 205, 339 Roth, T., 209 bounded repetition of, 210 Rothstein, J., 297 Toeplitz characterization, 170 Roy, C. L., 469 Reingold, E. M., 121 Rozenberg, G., 35, 36, 246, 340–342, 472, 477 Reiter, C. A., 426 Rudin, W., 121, 403, 473 Reitwiesner, G. W., 473 Rudin–Shapiro formal power series, 353 relation, 133 Rudin–Shapiro sequence, 78, 121, 122, 154, 171, 192, Rényi, A., 123 205, 339, 459, 462, 469 repeat-free, 32 2-kernel of, 186 repetitions in words, 14–16 complete, 459, 462 representation, 70 counterexamples and, 123 bijective, 101 generalized, 467, 479 Fibonacci, 105 palindromes in, 468 greedy, 71 square-root property, 79, 80, 121 unary, 73 Ruiz, J., 469 valid, 70 ruler function, 27 Le Rest, M., 38 run lengths, 25 Restivo, A., 30, 32, 33, 36, 472 running product, 196 Reutenauer, C., 36, 454, 472 running sum, 196, 206 Reverend Back’s abbey floor, 410, 426 Ruzsa, I. Z., 478 reversal, 5 Rytter, W., 35 reward, 5 Ryu, C. S., 469 ρ (spectral radius), 80 Ribenboim, P., 124 s-index, 30 Richards, J. I., 403 Saffari, B., 122, 123 Richomme, G., 35, 36, 296 Sajo, A., 342 Rideout, J., xv Sakarovitch, J., 125 Rider, D., 122, 123 Salejda, W., 468 Ridout, D., 404 Salem, R., 123, 472 Riemann zeta function, 360 Salemi, S., 30, 33 R (right part of an infinite word), 3 Salinas, S. R., 469 right dense, 345 Salman, A., 469 right-invariant, 133 Salomaa, A., 68, 125, 150, 151, 342 rightmost derivation, 144 Salon, O., 211, 376, 426 Rigo, M., 172 Samborski, J. R., 115, 473 Riklund, R., 469 Samuel, P., 68

568 Index

Sankar, P. V., 125 Shapiro, H. S., 121 Sano, Y., 123 Sharif, H., 376, 378, 426, 472, 477 Sanov, I. N., 34 Shechtman, D., 455, 467 Santean, N., 32 Shelton, R. O., 34–36 Santini-Bouchard, M. L., 343 Shen, C., 476 Sarkar, U. K., 210 Shepherd, S. C., 123 Sárközy, A., 120 shift map, 5, 27, 300 Saupe, D., 209 Shimshoni, M., 296 Schatz, J. A., 36, 37 Shiokawa, I., 120, 296, 342, 473 Scheicher, K., 454 Shiota, Y., 124 Schmidt’s theorem, 394–396 Shiu, P., 297 Schmidt, U., 37 Shiue, P. J.-S., 296, 297, 476 Schmidt, W. M., 68, 210, 394, 404, 472, 478 shshsh, 14 Schnellnhuber, H. J., 467 Shur, A. M., 33, 38 Schoenberg, I. J., 476 Shyr, H. J., 32, 471, 472 Schoenfeld, L., 69 Siddiqi, J. A., 123 Schrödingerequation, 467, 468 Sierpińskicarpet, 405, 406, 425, 426 tight-binding, 464 automaton for, 407 Schrödernumbers, 170 morphism for, 407 Schrödingeroperator, 464 Sierpiński,W., 426 Schützenberger, M. P., 10, 32, 38, 150, 171, 472 Siggia, E. D., 467 Scoppola, E., 468 k ,1 Scorer, R. S., 209 Silber, R., 476 Scott, D., 150 Silberger, D. M., 32, 471 Scott, W. T., 210 da Silva, C. A. A., 469 Scoville, R., 125 Simon, B., 468 Séébold, P., xv, 33, 36, 171, 208, 246, 296, 342, Simpson, R. J., 36, 68, 341, 472, 478 475, 476 singular continuous, 465 Seeman, T., 341 Siromoney, R., 36, 426 Segal, A. C., 116, 473 size, 9 Seki, S., 246 Skalba, M., 124, 126 Sekiguchi, T., 124 Skolem, T., 476 self-number, 121 Skordev, G., xv, 172, 211, 378, 426, 427, 477 Selfridge, J. L., 119, 472, 473 Skyum, S., 472 Sellers, J. A., 478 Slater, M., 120 Semenov, A. L., 350 Slater, N. B., 68 semigroup, 55 Slekys, A. G., 126 semiring, 56 Sloane, N. J. A., xiii, 118 Senechal, M., 467 Smeulders, A. W. M., 297 Senge, H. G., 120 Smith number, 121 Sentenac, P., 297 Smith, C. A. B., 209 separating collection, 117 Smith, H. J. S., 295, 296 sequence Smyth, W. F., 35, 37, 246, 343 digital, 83 Soittola, M., 151 infinite, 2 Solinas, J. A., 120 locally catenative, 237 Solitar, D., 34 morphic, 10, 212 Solomyak, B., 123 paperfolding, 181–185 solvable, 148 pure morphic, 10, 212 Songster, G. F., 125 substitutive, 245 Soni, R. P., 35, 36 Series, C., 296 Sós, V. T., 68, 126 Serrato, D., 210 space-filling curve, 155 set Spanier, E., 150 closed, 5 spectral radius, 80, 396, 398 k-automatic, 168 Speer, E. R., 124 open, 5 Spencer, Jr., S. M., 377 perfect, 24, 34 spins in Ising chain, 457 Severin, M., 469 Sprague, R., 476 Shallit, J. O., 32, 34, 67, 119, 123, 172, 209–211, 246, Sproull, R. F., 297 296, 341, 403, 426, 437, 454, 471–478 square, 14 Shamir, E., 150 abelian, 24 Shank, H., 171 minimal, 28, 31 Shannon, C. E., 125 square-root property Shapiro, H. N., 119–121 of the Rudin–Shapiro sequence, 79, 80, 121

Index 569

squarefree, 14–16, 24, 25, 27, 30, 32–35 substitutive sequence, 245 abelian, 24 subword, 2 strongly, 37 bispecial, 341 squarefree integer, 66 European use of, 31 squarefree morphism, 35 recurrent, 324 squarefree words subword complexity, 298, 341 algorithm for, 27 generalizations, 341 enumeration of, 35 in two dimensions, 413 lexicographically least, 30 succinctness, 130 uncountably many, 27 suffix, 2 squares Sukhanov, E. V., 37 efficient detection of, 35 sum of Fibonacci digits, 126 maximum number of, 29 sum-of-digits function, 74 start symbol, 143 sums of three squares, 206, 447 Steacy, R., 476 Sundar Raj, P. A., 297 Stearns, R. E., 403, 477 Surányi, J., 68 Steele, J. M., 295 Swart, D., 246 steering wheels, 471 Sweet, M. M., 172, 209, 210, 472, 477 Stein, A. H., 121 Swierczkowski,´ S., 68 Steinhardt, P. J., 467 symbolic trajectory, 340 Steinhau’s conjecture, 53 symbols, 1 Stephens, N. M., 475 symmetric, 133 Stepin, A. M., 343 syndetic, 346, 350 Stern–Brocot sequence, 478 Szabó,J., 126 Stewart, B. M., 120, 209 Szeg˝o,G., 404 Stewart, C., 120 Szüsz,P., 68, 476 Stewart, I., 471 stochastic matrix, 250 t (Thue–Morse infinite word), 15, 16, 23, 24, 26–29, Stockmeyer, L., 437, 478 31, 32, 36 Stockmeyer, P. K., 210 tables, 405 Stolarsky, K. B., 120, 124, 245, 295, 342, Taitslin, M. A., 36 473, 476 Tamura, J.-I., 209, 296, 297, 342, 426 Stolboushkin, A. P., 36 Tanaka, T.-A., 404 Stolfi, J., 426 Tang, C., 467 straight-line program, 242 Tang, S. C., 120 length, 242 Tao, R., 468, 469 Straus, E. G., 120 Tapsoba, T., 341 Strauss, N., 478 tartar, 471 strings, 1 Tassa, U., 295, 296, 342 strongly cubefree, 33 Taylor, W., 30, 38 strongly k-additive, 120 temperature, 457 strongly k-multiplicative, 120 Tenenbaum, G., 122, 124, 210, 282, 479 strongly nonrepetitive, 37 Terauchi, H., 468, 469 strongly primitive, 32 terminals, 143 strongly squarefree, 37 Terr, D. C., 121 Sturmian morphism, 296 Testard, D., 468 Sturmian sequences testes, 471 palindrome complexity of, 479 Tewari, U. B., 123 Sturmian words, 284 Thakur, D. S., 210, 373, 375–378 applications of, 295 ,63 computer graphics and, 296 Thierrin, G., 32, 472 repetitions in, 320 Thomas, A., 124 slope of, 284 Thompson, G. C., 473 subword complexity of, 312 three squares, sums of, 206, 447 total number of subwords in, 297 three-distance theorem, 53 Stux, I. E., 121 three-gap theorem, 53 sub-base for a topology, 6 Thue, A., 32, 208 submultiplicative, 251 Thue–Morse continued fraction, 394 Subramanian, K. G., 36, 426 Thue–Morse formal power series, 352 subsequence, 29, 31 as diagonal of rational function, 369 subset construction, 130 Thue–Morse morphism, 9, 28, 33, 339 subspace, 56 Thue–Morse partition, 208 substitution, 208 Thue–Morse real number, 387, 400, 401, 403 substitution matrix, 282 transcendence of, 387

570 Index

Thue–Morse sequence, 15, 23, 122, 152, 173, 192, Trigg, C. W., 115, 473 201, 203, 204, 206–208, 327, 336, 338–340, trivial morphism, 9 371, 425, 436 Troi, G., 121 appearance, 334 Trollope, J. R., 120, 124 automaton for, 174 Tromp, J., 341, 403, 476 chess and, 32 Trotter, W. T., 208 complexity of, 335 Tsai, C.-H., 469 generalized, 335 Tsai, S.-C., 473 critical exponent of, 24 Tu, F. K., 32 first differences, 201 Turban, L., 469 generalized, 208, 335 Turing machine, 146 history of, 32 Turing, A. M., 151, 379, 403 mirror invariance of, 29 Turner, J. C., 246, 476 music and, 33, 245, 448 Turnwald, G., 121 origins, 208 Twaddle, R. D., 125 physics and, 469 two-dimensional array, 405 polynomials and, 207 two-dimensional [k, l]-DFAO, 408 prefixes are primitive, 28 two-sided infinite words, 3, 218 recurrence function of, 329 periodic, 24, 220 recurrent subwords of, 325 two-sided, 26 Uchida, Y., 120 unusual infinite products and, 174 Ullman, J. D., 31, 150 Thue–Morse word, 15 ultimately periodic, 4 Thuswaldner, J. M., 120, 124, 125, 127, 473 subword complexity, 298 Tichy, R. F., 121, 123, 124, 454 ultrametric inequality, 7 Tijdeman, R., 341, 342 unambiguous grammar, 144 Toeplitz construction, 156 unambiguous numeration system, 71 Toeplitz word, 339 unary representation, 73 Toeplitz, O., 342 unbordered, 28 Toffin, P., 120 unexpended input, 145 Tognetti, K., 68 unfolding instructions, 182 Tompkins, C. B., 125 unidirectional infinite word, 3 Tong, P., 469 uniform, morphism, 9 topological space, 5 uniform tag sequences, 172 topology, 5 uniformly distributed, 49 Toshimitsu, T., 120 uniformly recurrent, 328 tower of Hanoi, 177 unique factorization domain, 66 cyclic, 210 unit, 59 more than three pegs, 209 unnormalized, 110 optimal solution, 178 unreachable state, 129 Town, H.-S., 472 Urbanek, F. J., 36, 210, 340 trace of a matrix, 457 Urzyczyn, P., 36 trace maps, 465 Uspensky, J. V., 295, 454, 476 Trakhtenbrot, B. A., 437 transcendence Vaidya, A. M., 121 of π, 43, 67 Vaillancourt, R., 123 of e, 43, 67 valid representation, 70 of characteristic real numbers, 393 valuation, 39 of Thue–Morse continued fraction, 394 Vanden Eynden, C., 401, 477 of Thue–Morse real number, 387 Vandermonde identity, 444, 449 transcendental, 41 Vandeth, D., xv, 37, 343 transducer Vanicek, T., 37 finite-state, 140 variable, 143 k-uniform, 140, 194 in a context-free grammar, 143 transfer matrix, 457 Varricchio, S., 36, 340, 341, 476, 477 transition diagram, 128 Vasiga, T. M. J., xv transition function, 128 vector transition matrix, 282 non-negative, 250 transitive, 133 positive, 250 tratratratra, 471 vector space, 56 Tresser, C., 476 Veech, W. A., 343 triangle inequality, 6 Venkov, B. A., 295, 296 triangular numbers, 449 Verraedt, R., 472 Tribonacci morphism, 246 Vielhaber, M., 201, 474

Index 571

Villemaire, R., 172, 350, 478 left-infinite, 3 Voss, K., 297 length of, 2 Vuillon, L., 342, 343, 426 primitive, 13, 32 right-infinite, 3 Wade, L. I., 361, 377 Toeplitz, 339 Wadel, L. B., 125 unbordered, 28 van der Waerden, B. L., 472 worm in Penrose tiling, 455 van der Waerden theorem, 207 Wozny, N., 342 Wakelin, J. H., 125 Wright, E. M., 67, 68, 172, 477 Wakulicz, A., 125 Wu, J.-S., 209, 210 Wall, H. S., 210 Wu, L.-D., 297 Walsh, T. R., 209, 210 Wyatt-Millington, C. W., 123 Wang, M.-w., xv, 32, 246, 471, 472, 475–477 Wythoff’s game, 293 Wantiez, P., 208 Wythoff, W. A., 476 wax, 187, 189, 219 Wayland, K., 121 Xie, H., 472 Weakley, W. D., 476 weakly cubefree, 33 Yaglom, A. M., 119, 454, 478 de Weger, B. M. M., 474 Yaglom, I. M., 119, 454, 478 Wegner, L., 426, 472 Yan, J. R., 469 Weiman, C., 297 Yao, J.-Y., xv, 208, 210, 363, 377 Wells, Jr., C. H., 125 Yarlagadda, R., 478 Wen, Z.-X., 209, 246 Yasutomi, S., 296 Wen, Z.-Y., 209, 246, 377, 404, 469 Yates, S., 121 Wender, P., 38 Yazdani, S., 208, 475 Weyl’s theorem, 50, 462 Yeung, D., xv Weyuker, E. J., 125 Yien, S.-C., 120 Wheeler, F. S., 473 You, J. Q., 469 White, T., 473 Yu, J., 377 width, 9 Yu, S., 31, 32 Wigderson, A., 437 Yu, S. S., 32 Wilansky, A., 121 Yu, X., 36 Wilf, H. S., 13, 32, 471, 473, 478 Yuen, C. K., 125 Wilson, D., xv, 113, 400, 475, 478 Winkler, P., 208 Zalcstein, Y., 35 Winklmann, K., 36 Zamboni, L. Q., xv, 246, 341–343, 404, Winley, G., 68 477 Wlazinski, F., 35 Zannier, U., 121 Wolfram, S., 404, 437 Zare, D., 479 Wood, D., 209, 246 Zariski, O., 68 Woodcock, C. F., 376, 378, 426, 472, 477 Zaslavsky, D., 469 Woods, D. R., 172 Zech, T., 34, 471 word chain, 242 Zeckendorf, E., 125 word graphs, 323, 343 Zeilberger, D., 35, 36 words, 1 zero pattern sequence, 78 bi-infinite, 31 Zetterström,H.-O., 473 binary balanced, 313 Zhong, J. X., 469 doubly infinite, 31 Zimin, A. I., 37 infinite, 2 Zohar, S., 125 one-sided, 3 Zorat, A., 297 two-sided, 3 Zygmund, A., 123