A161935 28-gonal numbers: a(n) = n*(13*n - 12).

0, 1, 28, 81, 160, 265, 396, 553, 736, 945, 1180, 1441, 1728, 2041, 2380, 2745, 3136, 3553, 3996, 4465, 4960, 5481, 6028, 6601, 7200, 7825, 8476, 9153, 9856, 10585, 11340, 12121, 12928, 13761, 14620, 15505, 16416, 17353, 18316, 19305, 20320, 21361, 22428, 23521

Offset: 0

Author

Pierre Gayet, Jun 22 2009

Keywords

• easy
• nonn

Comments

The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]

These are also the star 14-gonal numbers: a(n) = A051866(n) + 14*A000217(n-1). Luciano Ancora, Apr 04 2015

Examples

			G.f. = x + 28*x^2 + 81*x^3 + 160*x^4 + 265*x^5 + 396*x^6 + 553*x^7 + ...
		

Links

• Daniel Mondot, Table of n, a(n) for n = 0..1000
• Pierre Gayet, Note et Compte rendu (gif version).
• Pierre Gayet, Note et Compte Rendu (pdf version).
• Pierre Gayet, 98 séquences générées ... par la formule générale indiquée.
• Claude Monet, Nymphéas.
• Index to sequences related to polygonal numbers.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A051866, A161532, A161549, A161587, A161617, A162316.

Programs

Magma

[ (n+1)*(13*n+1): n in[0..50] ];

Mathematica

lst={}; Do[a=13*n^2+14*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst
Table[n*(13*n - 12), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

PARI

{a(n) = n*(13*n - 12)}; /* Michael Somos, Dec 07 2016 */

Formula

a(n+1) = a(n) + 26*n + 1. - Vincenzo Librandi, Nov 30 2010

a(n) = A000217(n) + 25*A000217(n-1). - Luciano Ancora, Apr 04 2015

Product_{n>=2} (1 - 1/a(n)) = 13/14. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 13*x^2). - Nikolaos Pantelidis, Feb 05 2023

From Elmo R. Oliveira, Dec 14 2024: (Start)

G.f.: x*(1 + 25*x)/(1 - x)^3.

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

Extensions

Edited by N. J. A. Sloane, Dec 07 2016 at the suggestion of Daniel Sterman.

Definition simplified by Omar E. Pol, Aug 10 2018