A161549 a(n) = 2*n^2 + 14*n + 1.

1, 17, 37, 61, 89, 121, 157, 197, 241, 289, 341, 397, 457, 521, 589, 661, 737, 817, 901, 989, 1081, 1177, 1277, 1381, 1489, 1601, 1717, 1837, 1961, 2089, 2221, 2357, 2497, 2641, 2789, 2941, 3097, 3257, 3421, 3589, 3761, 3937, 4117, 4301, 4489, 4681, 4877, 5077

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]

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, A161587, A161617, A161935, A162316.

Programs

Magma

[ 2*n^2+14*n+1: n in [0..50] ];

Mathematica

lst={}; Do[a=2*n^2+14*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 13 2009 *)
CoefficientList[Series[(1 + 14 x - 11 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Nov 08 2014 *)
Table[2n^2+14n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,17,37},50] (* Harvey P. Dale, Jul 14 2018 *)

PARI

Vec((1+14*x-11*x^2)/(1-x)^3 + O(x^100)) \\ Colin Barker, Nov 08 2014

Formula

a(n) = a(n-1) + 4*n + 12 (with a(0)=1). - Vincenzo Librandi, Nov 30 2010

G.f.: (1 + 14*x - 11*x^2)/(1-x)^3. - Vincenzo Librandi, Nov 08 2014

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

E.g.f.: (1 + 16*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