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

0,3

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

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).

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)

Example

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

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 *)

Prog

(Magma) [ (n+1)*(13*n+1): n in[0..50] ];
(PARI) {a(n) = n*(13*n - 12)}; /* Michael Somos, Dec 07 2016 */

Crossrefs

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

Sequence in context: A044547 A268733 A119178 * A378922 A039415 A043238

Adjacent sequences: A161932 A161933 A161934 * A161936 A161937 A161938

Keyword

easy,nonn,changed

Author

Pierre Gayet, Jun 22 2009

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

Status

approved