A069178 Centered 21-gonal numbers.

1, 22, 64, 127, 211, 316, 442, 589, 757, 946, 1156, 1387, 1639, 1912, 2206, 2521, 2857, 3214, 3592, 3991, 4411, 4852, 5314, 5797, 6301, 6826, 7372, 7939, 8527, 9136, 9766, 10417, 11089, 11782, 12496, 13231, 13987, 14764, 15562, 16381, 17221, 18082, 18964

Offset

1,2

Links

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

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) = (21n^2 - 21n + 2)/2

a(n) = 21*n + a(n-1) - 21 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010

G.f. -x*(1+19*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011

Binomial transform of [1, 21, 21, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 21, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011

a(n) = 1 + Sum_{i=1..n} 21*(i-1). - Wesley Ivan Hurt, May 25 2013

From Amiram Eldar, Jun 21 2020: (Start)

Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(13/21)*Pi/2)/sqrt(273).

Sum_{n>=1} a(n)/n! = 23*e/2 - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 23/(2*e) - 1. (End)

E.g.f.: exp(x)*(1 + 21*x^2/2)-1. - Nikolaos Pantelidis, Feb 06 2023

Mathematica

FoldList[#1 + #2 &, 1, 21 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 22, 64}, 60] (* Harvey P. Dale, Jun 13 2022 *)

Prog

(PARI) a(n)=(21*n^2-21*n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. centered polygonal numbers listed in A069190.

Sequence in context: A221595 A051874 A140390 * A081929 A305064 A124715

Adjacent sequences: A069175 A069176 A069177 * A069179 A069180 A069181

Keyword

easy,nonn

Author

Terrel Trotter, Jr., Apr 09 2002

Status

approved