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

1,2

Comments

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

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) = (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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A181149 A019063 A101740 * A124707 A126368 A125816

Adjacent sequences: A069123 A069124 A069125 * A069127 A069128 A069129

Keyword

easy,nice,nonn

Author

Terrel Trotter, Jr., Apr 07 2002

Status

approved