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
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).
Crossrefs
Cf. centered polygonal numbers listed in A069190.
Programs
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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 *)
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PARI
a(n)=(21*n^2-21*n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015
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