A069131 Centered 18-gonal numbers.

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, 17029, 17821

Offset: 1

Author

Terrel Trotter, Jr., Apr 07 2002

Keywords

• easy
• nice
• nonn

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)

Examples

			a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.
		

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.
• R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
• Index entries for sequences related to centered polygonal numbers.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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.

Programs

Magma

[9*n^2 - 9*n + 1 : n in [1..50]]; // Wesley Ivan Hurt, May 05 2021

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 *)

PARI

a(n)=9*n^2-9*n+1 \\ Charles R Greathouse IV, Oct 07 2015

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