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: Polygonal number, Pyramidal number, 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 figurate number of rank n where generated polygon is of p sides ja ln = Jacobsthal Lucas number 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 = Pronic number c a r ln = Carol number m e rn = Mersenne number, where n is prime c u ln = Cullen number T h a n = Thabit ibn kurrah number won = Woodall number II. INTERESTING RELATIONS N 4 1. t2 0 ,n98 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 ] nn11 NN 2 98nn nn11 NNNNN( 1)( 2 1) ( 1) 98 22 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 12 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 12 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 N 2 4 2 9nn 8 t2 0 ,n98 p N t 3, N n 1 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 16 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 16 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 42ttnn42 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,nn3 t 2 0 , 2 8 n 2 7 n 2 7 n 2 4 n 2 t5 8,nn33 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 ,n3 c t 3, n 1 5 n 2 g n Proof 2 t2 0 ,nn3 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 ,n3 ct 3, n 1 5 n 2 g n 6 n 52 f) t2 t 92 2 p 2 4 n 2 0 ,n 3, n n n Proof 4 3 2 t2 0 ,nn2 t 3, 9 n n 8 n 2 2 2 2 3 2 9 n n n n 1 8 n 5 2 2 t2 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,n4 g n 3 cp n 2 cp n Proof 3 n t20,nn4 g 9 n 4 n 33 7n 4 n 2 n 1 4 6 n t20,n4 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 ,n1 s n t 8 , n Proof 2 t20,n 98 n n 22 6n 6 n 1 3 n 2 n 1 t2 0 ,n1 s n t 8 , n 1 0 1 0 4 17. n t2 0 ,n1 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 ,n1 t 2 4 , n 6 p n 3 cp n 6 cp n 18.