A161532 a(n) = 2*n^2 + 8*n + 1.

1, 11, 25, 43, 65, 91, 121, 155, 193, 235, 281, 331, 385, 443, 505, 571, 641, 715, 793, 875, 961, 1051, 1145, 1243, 1345, 1451, 1561, 1675, 1793, 1915, 2041, 2171, 2305, 2443, 2585, 2731, 2881, 3035, 3193, 3355, 3521, 3691, 3865, 4043, 4225, 4411, 4601, 4795, 4993

Offset: 0

Author

Pierre Gayet, Jun 13 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]

Numbers of the form 2*n^2 - 7. - Boris Putievskiy, Feb 04 2013

Links

• Pierre Gayet, Table of n, a(n) for n = 0..10000
• 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. A161549, A161587, A161617, A161935, A162316.

Programs

Magma

[ 2*n^2+8*n+1: n in [0..50] ];

Mathematica

Table[2*n^2+8*n+1, {n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2009 *)
CoefficientList[Series[(1 + 8*x - 5*x^2)/(1-x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Feb 07 2013 *)

PARI

a(n)=2*n^2+8*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = a(n-1) + 4*n + 6 (with a(0)=1). - Vincenzo Librandi, Nov 30 2010

G.f.: (1 + 8*x - 5*x^2)/(1 - x)^3. - Vincenzo Librandi, Feb 07 2013

From Elmo R. Oliveira, Oct 27 2024: (Start)

E.g.f.: (1 + 10*x + 2*x^2)*exp(x).

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

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Jun 13 2009