A162316 a(n) = 5*n^2 + 20*n + 1.

1, 26, 61, 106, 161, 226, 301, 386, 481, 586, 701, 826, 961, 1106, 1261, 1426, 1601, 1786, 1981, 2186, 2401, 2626, 2861, 3106, 3361, 3626, 3901, 4186, 4481, 4786, 5101, 5426, 5761, 6106, 6461, 6826, 7201, 7586, 7981, 8386, 8801, 9226, 9661, 10106, 10561, 11026

Offset: 0

Author

Pierre Gayet, Jul 01 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]

Links

• Pierre Gayet, Table of n, a(n) for n = 0..9999
• 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 entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A161532, A161549, A161587, A161617, A161935.

Programs

Magma

[ 5*n^2+20*n+1: n in [0..50] ];

Mathematica

lst={}; Do[a=5*n^2+20*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst
Table[5n^2+20n+1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,26,61},40] (* or *) CoefficientList[Series[(14x^2-23x-1)/(x-1)^3,{x,0,40}],x] (* Harvey P. Dale, May 07 2023 *)

PARI

a(n)=5*n^2+20*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = a(n-1) + 10*n + 15 (with a(0)=1). - Vincenzo Librandi, Dec 02 2010

G.f.: (14*x^2 - 23*x - 1)/(x - 1)^3. - Harvey P. Dale, May 07 2023

From Elmo R. Oliveira, Oct 25 2024: (Start)

E.g.f.: (5*x^2 + 25*x + 1)*exp(x).

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