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Sequences from Hexagonal Pyramid of Integers International Mathematical Forum, Vol. 6, 2011, no. 17, 821 - 827 Sequences from Hexagonal Pyramid of Integers T. Aaron Gulliver Department of Electrical and Computer Engineering University of Victoria, P.O.Box 3055, STN CSC Victoria, BC, V8W 3P6, Canada [email protected] Abstract This paper presents a number of sequences based on integers ar- ranged in a hexagonal pyramid structure. This approach provides a simple derivation of some well known sequences. In addition, a number of new integer sequences are obtained. Mathematics Subject Classification: 11Y55 Keywords: integer arrays, integer sequences 1. Introduction Previously, several well-known sequences (and many new sequences), were derived from tetrahedral (three-sided) [2], square (four-sided) [3], and pentag- onal (five-sided) [5] pyramids of integers. For example, the number of elements in the square pyramid is n 2 2 2 2 2 2 2 1 sn =1 +2 +3 +4 +5 + ...+ n = i = n(n + 1)(2n +1), (1) i=1 6 where n is the height of the pyramid. Starting from n = 1, we have 1, 5, 14, 30, 55,... (2) which is sequence A000330 in the Encyclopedia of Integer Sequences main- tained by Sloane [6], and appropriately called the square pyramidal numbers. Sequences based on a hexagonal pyramid are given in the next section. 822 T. Aaron Gulliver 2. Hexagonal Pyramids of Integers A hexagonal pyramidal array of integers has a structure with 1 at the top, 2 to 7 on the second level, 7 to 22 on the third level, etc. An illustration of the fourth level is give in Fig. 1. The number of elements on level i is a Figure 1: The fourth level of the hexagonal pyramid of integers. hexagonal number given by i(2i − 1) and the resulting integer sequence is si =1, 6, 15, 28, 45,... The number of elements in the pyramid is then n 1 i(2i − 1) = n(n + 1)(4n − 1), (3) i=1 6 where n is the height of the pyramid. Starting from n = 1, we have 1, 7, 22, 50, 95,... (4) which is sequence A002412 in [6], and are called the hexagonal pyramidal numbers. A number of new sequences can be obtained from this structure, depending on the arrangement of numbers on a level. In this paper, we consider the following arrangement. For the top two levels, this is 2 37 1 , 46 5 Sequences from hexagonal pyramid of integers 823 For the third level, we have 8 913 14 10 12 22 15 11 21 . 16 20 17 19 18 In addition to (4), the following simple sequences are obtained from the integers on the corners of the pyramid 1, 2, 8, 23, 51, ... 1, 3, 14, 38, 79, ... 1, 4, 16, 41, 83, ... (5) 1, 5, 18, 44, 87, ... 1, 6, 20, 47, 91, ... The first of these is just (4) + 1 and is given by 1 3 2 sn = (4n +3n − n +6). 6 The second sequence is new and is given by 1 3 2 sn = (4n +3n − 25n + 24), 6 while the third is generated by 1 3 2 sn = (4n +3n − 19n + 18). 6 In general, the second through fifth sequences in (5) are given by 1 3 2 sn = (4n +3n − 25n + 24) + l(n − 1) 6 for l = 0 to 3. For example, the last sequence is generated by 1 2 sn = (n + 2)(4n − 5n +3). 6 and (4) is obtained with l =4. 824 T. Aaron Gulliver The sum of the elements on the bottom rows of the pyramid (starting from the top and moving clockwise), give the sequences 1, 5, 31, 114, 305, ... 1, 7, 45, 158, 405, ... 1, 9, 51, 170, 425, ... (6) 1, 11, 57, 182, 445, ... 1, 13, 63, 194, 465, ... 1, 9, 43, 138, 345, ... The first sequence is generated by 1 3 2 sn = n(4n − 5n − 4n + 11) 6 the last is obtained from 1 3 2 sn = n(4n − 5n +8n − 1) 6 while the remainder are given by 1 3 2 sn = n(4n +3n − 22n +21+6l(n − 1)) 6 for l = 0 to 3. Now consider rays in the pyramid towards the corners, starting from the smallest integer on a level. The sum of the elements in the leftmost ray is the first sequence in (6). The next ray gives the sequence 1, 6, 34, 120, 315, ... with terms 1 2 sn = n(n + 1)(4n − 9n +8). 6 In general, these sequences are generated by 1 3 2 sn = n(4n − 5n +(3l − 4)n − 3l + 11). 6 for l = 0 to 3, so the remaining sequences are 1, 7, 37, 126, 325, ... 1, 8, 40, 132, 335, ... Now consider wedges in the pyramid. The sum of the elements in the leftmost wedge results in the sequence 1, 9, 72, 320, 1005, ... Sequences from hexagonal pyramid of integers 825 with terms 1 3 2 sn = n(n + 1)(4n − 3n − 7n + 12). 12 The next wedge gives the sequence 1, 11, 80, 340, 1045, ... with terms 1 3 2 sn = n(n + 1)(4n − 3n − 3n +8). 12 In general, the wedges are given by 1 3 2 sn = n(n + 1)(4n − 3n +(4l − 7)n − 4l + 12). 12 for l = 0 to 3, so the remaining sequences are 1, 13, 88, 360, 1085, ... 1, 15, 96, 380, 1125, ... Combining the first two wedges gives 1, 14, 118, 540, 1735, ... which is generated by 1 3 2 sn = n(n + 1)(4n − 7n +4n +2), 6 and adding the next wedge gives 1, 20, 169, 774, 2495, ... with 1 4 3 2 sn = n(12n − 13n +2n +13n − 2). 12 Finally, combining the last wedge gives the sum of the elements in each level 1, 27, 225, 1022, 3285, ... with 1 3 2 sn = n(2n − 1)(4n − 3n +2n +3). 6 826 T. Aaron Gulliver Now adding the elements on all the levels gives n 1 3 2 sn = i(2i − 1)(4i − 3i +2i +3) i=1 6 (7) 1 = n(n + 1)(4n − 1)(4n3 +3n2 − n +6) 72 which gives 1, 28, 253, 1275, 4560, ... (8) This result can also be obtain by summing the positive integers up to the values in (4) n(n+1)(4n−1)/6 sn = i i=1 (9) 1 = n(n + 1)(4n − 1)(4n3 +3n2 − n +6). 72 In general, the wedge values are given by 1 4 3 2 2 sn = n (4l +4)n +(1− 7l)n +(2l +2l − 10)n 12 +(5 − 3l2 +10l)n + l2 − 9l + 12) for l = 0 to 3. Summing these values provides the partial wedge sums n 1 4 3 2 2 2 sn = l (4l +4)i +(1− 7l)i +(2l +2l − 10)i +(5− 3l +10l)i 12 i=1 +l2 − 9l + 12) 1 = n(n +1) (20 + 20l)n4 + (46 − 2l)n3 + (15l2 − 38l − 56)n2 360 +(98l − 15l2 − 34)n − 78l + 204 (10) which for l = 0 to 2 is 1, 10, 82, 402, 1407, ... 1, 15, 133, 673, 2408, ... 1, 21, 190, 964, 3459, ... and (8) for l =3. Sequences from hexagonal pyramid of integers 827 References [1] T.A. Gulliver, Sequences from Arrays of Integers, Int. Math. J. 1 323–332 (2002). [2] T.A. Gulliver, Sequences from Integer Tetrahedrons, Int. Math. Forum, 1, 517–521 (2006). [3] T.A. Gulliver, Sequences from Pyramids of Integers, Int. J. Pure and Ap- plied Math. 36 161–165, (2007). [4] T.A. Gulliver, Sequences from Cubes of Integers, Int. Math. J. 4, 439–445, (2003). Correction Int. Math. Forum, vol. 1, no, 11, pp. 523-524. [5] T.A. Gulliver, Sequences from Pentagonal Pyramids of Integers [6] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/˜njas/sequences/index.html. Received: November, 2009.
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