A069173 Centered 22-gonal numbers.

1, 23, 67, 133, 221, 331, 463, 617, 793, 991, 1211, 1453, 1717, 2003, 2311, 2641, 2993, 3367, 3763, 4181, 4621, 5083, 5567, 6073, 6601, 7151, 7723, 8317, 8933, 9571, 10231, 10913, 11617, 12343, 13091, 13861, 14653, 15467, 16303, 17161, 18041, 18943, 19867, 20813

Offset: 1

Author

Terrel Trotter, Jr., Apr 09 2002

Keywords

• easy
• nonn

Examples

			a(5) = 221 because 11*5^2 - 11*5 + 1 = 275 - 55 + 1 = 221
For n=2, a(2)=22*2+1-22=23; n=3, a(3)=22*3+23-22=67; n=4, a(4)=22*4+67-22=133.
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• 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.

Programs

Mathematica

FoldList[#1 + #2 &, 1, 22 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)

PARI

a(n)=11*n^2-11*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 11*n^2 - 11*n + 1.

a(n) = 22*n + a(n-1) - 22 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010

From Amiram Eldar, Jun 21 2020: (Start)

Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(7/11)*Pi/2)/sqrt(77).

Sum_{n>=1} a(n)/n! = 12*e - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 12/e - 1. (End)

E.g.f.: exp(x)*(1 + 11 * x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

From Elmo R. Oliveira, Oct 22 2024: (Start)

G.f.: x*(1 + 20*x + x^2)/(1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)