A069126 Centered 13-gonal numbers.

1, 14, 40, 79, 131, 196, 274, 365, 469, 586, 716, 859, 1015, 1184, 1366, 1561, 1769, 1990, 2224, 2471, 2731, 3004, 3290, 3589, 3901, 4226, 4564, 4915, 5279, 5656, 6046, 6449, 6865, 7294, 7736, 8191, 8659, 9140, 9634, 10141, 10661, 11194

Offset: 1

Author

Terrel Trotter, Jr., Apr 07 2002

Keywords

• easy
• nice
• nonn

Comments

Centered tridecagonal numbers or centered triskaidecagonal numbers. - Omar E. Pol, Oct 03 2011

Examples

			a(5) = 131 because 131 = (13*5^2 - 13*5 + 2)/2 = (325 - 65 + 2)/2 = 262/2 = 131.
		

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. A001263, A001844, A003215, A005448, A005891, A069099, A152741.

Programs

Mathematica

FoldList[#1 + #2 &, 1, 13 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,14,40},60] (* Harvey P. Dale, Jan 20 2014 *)
With[{nn=50},Total/@Thread[{PolygonalNumber[13,Range[nn]],Range[0,nn-1]^2}]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Aug 29 2016 *)

PARI

a(n)=13*n(n-1)/2+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (13n^2 - 13n + 2)/2.

Binomial transform of [1, 13, 13, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 13, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007

a(n) = 13*n+a(n-1)-13 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010

G.f.: -x*(1+11*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011

a(n) = A152741(n-1) + 1. - Omar E. Pol, Oct 03 2011

From Amiram Eldar, Jun 21 2020: (Start)

Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(5/13)*Pi/2)/sqrt(65).

Sum_{n>=1} a(n)/n! = 15*e/2 - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 15/(2*e) - 1. (End)

E.g.f.: exp(x)*(1 + 13*x^2/2) - 1. - Stefano Spezia, May 15 2022