A080859 a(n) = 6*n^2 + 4*n + 1.

1, 11, 33, 67, 113, 171, 241, 323, 417, 523, 641, 771, 913, 1067, 1233, 1411, 1601, 1803, 2017, 2243, 2481, 2731, 2993, 3267, 3553, 3851, 4161, 4483, 4817, 5163, 5521, 5891, 6273, 6667, 7073, 7491, 7921, 8363, 8817, 9283, 9761, 10251, 10753, 11267

Offset: 0

Author

Paul Barry, Feb 23 2003

Keywords

• nonn
• easy

Comments

The old definition of this sequence was "Generalized polygonal numbers".

Column T(n,4) of A080853.

Sequence found by reading the line from 1, in the direction 1, 11, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Subsequence of A186424.

Cf. A000384, A001318, A033579, A033581.

Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.

Programs

Magma

[6*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Apr 29 2016

Mathematica

Table[6 n^2 + 4 n + 1, {n, 0, 50}] (* Vincenzo Librandi, Apr 29 2016 *)

PARI

a(n)=6*n^2+4*n+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (C(3,0) + (C(5,2) - 2)*x + C(3,2)*x^2)/(1-x)^3 = (1 + 8*x + 3*x^2)/(1-x)^3.

E.g.f.: (1 + 10*x + 6*x^2)*exp(x). - Vincenzo Librandi, Apr 29 2016

a(n) = C(4,0) + C(4,1)n + C(4,2)n^2.

a(n) = A186424(2*n).

a(n) = 12*n + a(n-1) - 2 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

a(n) = (n+1)*A000384(n+1) - n*A000384(n). - Bruno Berselli, Dec 10 2012

a(n) = (n+1)^4 mod n^3 for n >= 7. - J. M. Bergot, Aug 14 2017

a(n) = (2*n+1)^2 + 2*n^2. - Robert FERREOL, Jan 13 2024

Extensions

Definition replaced with the closed form by Bruno Berselli, Dec 10 2012