International Journal of Computational Engineering Research||Vol, 03||Issue, 5||

Observations on Icosagonal number M.A.Gopalan1, Manju Somanath2, K.Geetha3 1. Department of Mathematics, Shrimati Indira Gandhi College, Trichirappalli, 2. Department of Mathematics, National College, Trichirappalli, 3. Department of Mathematics, Cauvery College for Women, Trichirapalli,

ABSTRACT We obtain different relations among Icosagonal number and other two, three and four dimensional figurate numbers.

Keyword: , , centered polygonal number, Centered pyramidal number, Special number

I. INTRODUCTION The numbers that can be represented by a regular arrangement of points are called the polygonal numbers (also known as two dimensional figurate numbers). The polygonal number series can be summed to form solid three dimensional figurate numbers called Pyramidal numbers that be illustrated by pyramids [1].Numbers have varieties of patterns [2-16] and varieties of range and richness. In this communication we deal 2 with Icosagonal numbers given by t20,n 98 n n and various interesting relations among these numbers are exhibited by means of theorems involving the relations.

Notation t mn, = Polygonal number of rank n with sides m m p n = Pyramidal number of rank n with sides m

Fm,, n p = m-dimensional of rank n where generated polygon is of p sides

ja ln = Jacobsthal ctmn, = Centered Polygonal number of rank n with sides m m cpn = Centered Pyramidal number of rank n with sides m g n = Gnomonic number of rank n with sides m p n = c a r ln = m e rn = Mersenne number, where n is prime c u ln =

T h a n = Thabit ibn kurrah number won = Woodall number

II. INTERESTING RELATIONS N 4 1.  t2 0 ,n98 p N t 3, N n 1 Proof

www.ijceronline.com ||May ||2013|| Page 28 International Journal of Computational Engineering Research||Vol, 03||Issue, 5||

NN 2 t20,n [9 n 8 n ] nn11 NN 2 98nn nn11 NNNNN( 1)( 2  1) (  1) 98 22 N 4  t2 0 ,n98 p N t 3, N n 1 65 2.  t20,n t 12, n 4 t 7, n  t 11, n  2 5 cp n  2 2 p n Proof 32  t20,n t 12, n 4 t 7, n  t 11, n    1 4 n  1 1 n

3 3 2  2 5n  1 1  n  n 

65  t20,n t 12, n 4 t 7, n  t 11, n  2 5 cp n  2 2 p n

3. t2 0 ,n 12 t 1 1, n  g n  1 5 n  2 Proof 2 t2 0 ,n  1 9 n  1 0 n  1 2 9n  7 n  2 n  1  1 5 n  2

2t1 1, nn  g  1 5 n  2

t2 0 ,n 12 t 1 1, n  g n  1 5 n  2

4. t20,n 1 t 20, n 1  n  t 38, n  18 Proof 2 t2 0 ,nn 1 t 2 0 , 1 1 8 n  1 6 n  1 8 2 1 8n  1 7 n  n  1 8

t20,n 1 t 20, n 1  n  t 38, n  18

5. The following triples are in arithmetic progression

a)  t1 9 ,n,, t 2 0 , n t 2 1, n 

Proof

1 22 t1 9 ,nn t 2 1, 1 7 n  1 5 n  1 9 n  1 7 n  2 2 2 9nn 8 

t1 9 ,n t 2 1, n2 t 2 0 , n

b)  t2 0 ,n,, t 2 1, n t 2 2 , n  www.ijceronline.com ||May ||2013|| Page 29 International Journal of Computational Engineering Research||Vol, 03||Issue, 5||

Proof 22 t2 0 ,nn t 2 2 , 9 n  8 n  1 0 n  9 n

t2 0 ,n t 2 2 , n2 t 2 1, n

c)  t1 0 ,n,, t 2 0 , n t 3 0 , n  Proof 22 t1 0 ,nn t 3 0 , 4 n  3 n  1 4 n  1 3 n

t1 0 ,n t 3 0 , n2 t 2 0 , n

d)  t1 8,nnn,, t 2 0 , t 2 2 ,  Proof 22 t1 8,nn t 2 2 , 8 n  7 n  1 0 n  9 n

t1 8 ,n t 2 2 , n2 t 2 0 , n

e)  t1 6 ,nnn,, t 2 0 , t 2 4 ,  Proof 22 t1 6 ,nn t 2 4 , 7 n  6 n  1 1 n  1 0 n

t1 6 ,n t 2 4 , n2 t 2 0 , n 1 0 5 6. n t2 0 ,n 16 p n  2 p n  1 2 t 3, n Proof 32 n t2 0 ,n  1 9 n  1 0 n  n 3 2 3 2 2 8n  3 n  5 n  n  n  6 n  6 n 1 0 5 n t2 0 ,n 16 p n  2 p n  1 2 t 3, n

7. tn 1 7  9 ky  2 6 m er 2 0 , 2 nn Proof 2 nn t n 9 2 8 2 2 0 ,2    

2 n n n 92  22   1  262  117 

tn 1 7  9 ky  2 6 m er 2 0 , 2 nn

8. tnn t  m er  ca rl  2 2 0 , 2 1 6 , 2 2 nn Proof 2 nn ttnn 2 2  2 2 2 0 , 2 1 6 , 2    

2n 2 n n  1  2  1  2  2  1  2

tnn t  m er  ca rl  2 2 0 , 2 1 6 , 2 2 nn

9. n3 tnn t  cu l  w o  8, 2 2 0 , 2  nn Proof www.ijceronline.com ||May ||2013|| Page 30 International Journal of Computational Engineering Research||Vol, 03||Issue, 5||

nn n3 tnn t  n 2  1  n 2  1  8, 2 2 0 , 2     

n3 tnn t  cu l  w o  8, 2 2 0 , 2  nn

10. 31tnn t  T h a  8, 2 2 0 , 2 2 n Proof 2 n 3ttnn 3 2 8,2 2 0 ,2  

T h a 2 n 1

11. 4 2tnn t  ja l  1  1 2 , 2 2 0 , 2  22n  Proof 2 n 42ttnn42  1 2 , 2 2 0 , 2   

4 2tnn t  ja l  1  1 2 , 2 2 0 , 2  22n  12. The following is a Nasty number

a) t2 0 ,nn t 8 , 6 n Proof 22 t2 0 ,nn t 8, 9 n  8 n  3 n  2 n 2 66nn

b) 6 t5 8,nn 3 t 2 0 , 3 n  Proof 22 t5 8,nn3 t 2 0 ,  2 8 n  2 7 n  2 7 n  2 4 n 2 t5 8,nn33 t 2 0 ,  n  n

t2 0 ,nn t 1 6 , n c) 6  t 6,n

Proof 2 t2 0 ,nn t 1 6 , n 2 nn  2 t6,n 2 nn

d) 6  t8,n t 2 4 , n t 2 0 , n  Proof 22 t8,n t 2 4 , n  t 2 0 , n 3 n  2 n  2 n  2 n 2  n

e) t2 0 ,n3 c t 3, n  1 5 n  2 g n Proof 2 t2 0 ,nn3 ct 3,  6 n  1 5 n  4 n  2

www.ijceronline.com ||May ||2013|| Page 31 International Journal of Computational Engineering Research||Vol, 03||Issue, 5||

2 6n  15 n  2(2 n  1)

2 t2 0 ,n3 ct 3, n  1 5 n  2 g n  6 n 52 f) t2 t  92  2 p  2 4 n  2 0 ,n 3, n n n Proof 4 3 2 t2 0 ,nn2 t 3,  9 n  n  8 n 2 2 2 2 3 2 9 n  n  n  n  1 8 n  5 2 2 t2 t  92  2 p  2 4 n  6 n  2 0 ,n 3, n n n 7 13. t t 7 n  3 6 t2  3 0 p  2 t  2 0 ,n 6 , n 3,n n 7 , n Proof 4 3 2  t2 0 ,nn t 6 ,  1 8 n  2 5 n  8 n

4 2 3 2 2 1 8 n  n  5 5 n  3 n  2 n  5 n  3 n  7 n

7 t t 7 n  3 6 t2  3 0 p  2 t  2 0 ,n 6 , n 3,n n 7 , n

1 4 6 14. n t20,n4 g n  3 cp n  2 cp n Proof 3 n t20,nn4 g  9 n  4 n

33 7n  4 n  2 n 1 4 6 n t20,n4 g n  3 cp n  2 cp n

15. t20,n t 14, n  t 24, n  t 18, n

Proof 2 t2 0 ,nn t 1 4 , 33 n  n 2 t2 4 ,nn t 1 8, 33 n  n

t20,n t 14, n  t 24, n  t 18, n

16. t2 0 ,n1  s n  t 8 , n Proof 2 t20,n 98 n n

22  6n  6 n  1  3 n  2 n   1

t2 0 ,n1  s n  t 8 , n 1 0 1 0 4 17. n t2 0 ,n1  t 2 4 , n  6 p n  3 cp n  6 cp n Proof 32 n t2 0 ,nn t 2 4 , 9 n  2 n  1 0 n www.ijceronline.com ||May ||2013|| Page 32 International Journal of Computational Engineering Research||Vol, 03||Issue, 5||

3 2 3 3  8n  3 n  5 n  5 n  2 n  2 2 n  n  n

1 0 1 0 4 n t2 0 ,n1  t 2 4 , n  6 p n  3 cp n  6 cp n

18. 52 n18 pn t20, n  17 n  108 F 4, n ,4 Proof 5 4 2 n1 8 pnn t20,   9 n  8 n

4 2 2 9 n  n  17 n

52 n18 pn t20, n  17 n  108 F 4, n ,4

1 4 6 19. t2 0 ,n p n 36 n  p n  t 3 4 , n Proof 32 1 4 2 43n n n t20,nn p 98 n  n  2 3 2 2  4n  3 n  n  1 6 n  1 5 n  3 n

1 4 6 t2 0 ,n p n 36 n  p n  t 3 4 , n

20. 4 t20,n t 38, n  t 18, n  t 22, n  n

Proof

2 t2 0 ,nn t 3 8 , n (1)

2 t2 0 ,n t 1 8 , n t 2 2 , n (2) Add (1) and (2), we get

4 t20,n t 38, n  t 18, n  t 22, n  n

21. t5 8 ,n t 4 0 , n  n  t 2 0 , n Proof 2 t5 8,nn t 4 0 , 99 n  n

tn20,n

t5 8 ,n t 4 0 , n  n  t 2 0 , n

18 22.  t2 0 ,n g n 46 n  cp n  t 5 2 , n

Proof 32  t20,nn g 1 8 n  2 5 n  8 n

32 6 3n  4 n  2 5 n  2 4 n  4 n

18  t2 0 ,n g n 46 n  cp n  t 5 2 , n

www.ijceronline.com ||May ||2013|| Page 33 International Journal of Computational Engineering Research||Vol, 03||Issue, 5||

REFERENCES [1.] Beiler A. H., Ch.18 in Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, New York, Dover, (1996), 184-199. [2.] Bert Miller, Nasty Numbers, The Mathematics Teachers, 73(9), (1980), 649. [3.] Bhatia B.L., and Mohanty Supriya, Nasty Numbers and their characterization, Mathematical Education, I (1), (1985), 34-37. [4.] Bhanu Murthy T.S., Ancient Indian Mathematics, New Age International Publishers Limited, New Delhi, (1990). [5.] Conway J.H., and Guy R.K., The Book of Numbers, New York Springer – Verlag, (1996), 44-48. [6.] Dickson L.E., History of the Numbers, Chelsea Publishing Company, New York, (1952). [7.] Meyyappan M., Ramanujan Numbers, S.Chand and Company Ltd., First Edition, (1996). [8.] Horadam A.F., Jacobsthal Representation Numbers, Fib. Quart., 34,(1996), 40-54. [9.] Shailesh Shirali, Primer on Number Sequences, Mathematics Student, 45(2), (1997), 63-73. [10.] Gopalan M.A., and Devibala S., “On Lanekal Number”, Antarctica J. Math, 4(1), (2007), 35-39. [11.] Gopalan M.A., Manju Somanath, and Vanitha N., “A Remarkable Lanekel Sequence”, Proc. Nat. Acad. Sci. India 77(A), II(2007), 139-142. [12.] Gopalan M.A., Manju Somanath, and Vanitha N., “On R2 Numbers”, Acta Ciencia Indica, XXXIIIM(2), (2007), 617- 619. [13.] Gopalan M.A., and Gnanam A., “Star Numbers”, Math. Sci. Res.J., 12(12), (2008), 303-308. [14.] Gopalan M.A., and Gnanam A., “A Notable Sequence”, Math. Sci.Res. J., 1(1) (2008) 7-15. [15.] Gopalan M.A., and Gnanam A., “Magna Numbers”, Indian Journal of Mathematical Sciences, 5(1), (2009), 33-34. [16.] Gopalan M.A., and Gnanam A., “Four dimensional pyramidal numbers”, Pacific Asian Journal of Mathematics, Vol- 4, No-1, Jan-June, (2010), 53-62.

www.ijceronline.com ||May ||2013|| Page 34