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 A069131 Centered 18-gonal numbers. 10
 1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010 Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011 From Lamine Ngom, Aug 19 2021: (Start) Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section). a(n) is a bisection of A195042. a(n) is a trisection of A028387. a(n) + 1 is promic (A002378). a(n) + 2 is a trisection of A002061. a(n) + 9 is the arithmetic mean of its neighbors. 4*a(n) + 5 is a square: A016945(n)^2. (End) LINKS Ivan Panchenko, Table of n, a(n) for n = 1..1000 John Elias, Illustration of Initial Terms: Triangular & Hexagonal Configurations Lamine Ngom, An origin of A069131 (illustration) Leo Tavares, Illustration: Tri-Hexagons Eric Weisstein's World of Mathematics, Centered Polygonal Numbers Index entries for sequences related to centered polygonal numbers Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 9*n^2 - 9*n + 1. a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010 G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014 From Amiram Eldar, Jun 21 2020: (Start) Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)). Sum_{n>=1} a(n)/n! = 10*e - 1. Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End) From Lamine Ngom, Aug 19 2021: (Start) a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1. a(n) = 3*A003215(n) - 2. a(n) = A247792(n) - 9*n. a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0. a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0. a(n+1) - a(n) = 18*n = A008600(n). (End) From Leo Tavares, Oct 31 2021: (Start) a(n)= A000290(n) + A139278(n-1) a(n) = A069129(n) + A002378(n-1) a(n) = A062786(n) + 8*A000217(n-1) a(n) = A062786(n) + A033996(n-1) a(n) = A060544(n) + 9*A000217(n-1) a(n) = A060544(n) + A027468(n-1) a(n) = A016754(n-1) + 10*A000217(n-1) a(n) = A016754(n-1) + A124080 a(n) = A069099(n) + 11*A000217(n-1) a(n) = A069099(n) + A152740(n-1) a(n) = A003215(n-1) + 12*A000217(n-1) a(n) = A003215(n-1) + A049598(n-1) a(n) = A005891(n-1) + 13*A000217(n-1) a(n) = A005891(n-1) + A152741(n-1) a(n) = A001844(n) + 14*A000217(n-1) a(n) = A001844(n) + A163756(n-1) a(n) = A005448(n) + 15*A000217(n-1) a(n) = A005448(n) + A194715(n-1). (End) E.g.f.: exp(x)*(1 + 9*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023 EXAMPLE a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181. MATHEMATICA FoldList[#1 + #2 &, 1, 18 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *) LinearRecurrence[{3, -3, 1}, {1, 19, 55}, 50] (* Harvey P. Dale, Jan 20 2014 *) PROG (PARI) a(n)=9*n^2-9*n+1 \\ Charles R Greathouse IV, Oct 07 2015 (Magma) [9*n^2 - 9*n + 1 : n in [1..50]]; // Wesley Ivan Hurt, May 05 2021 CROSSREFS Cf. centered polygonal numbers listed in A069190. Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061. Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715. Sequence in context: A371056 A044121 A044502 * A124712 A126373 A125818 Adjacent sequences: A069128 A069129 A069130 * A069132 A069133 A069134 KEYWORD easy,nice,nonn AUTHOR Terrel Trotter, Jr., Apr 07 2002 STATUS approved

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Last modified July 14 00:54 EDT 2024. Contains 374290 sequences. (Running on oeis4.)