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

0,2

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).

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

Mathematica

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

Prog

(PARI) a(n)=6*n^2+4*n+1 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [6*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Apr 29 2016

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.

Sequence in context: A249166 A296543 A152740 * A063036 A163673 A212132

Adjacent sequences: A080856 A080857 A080858 * A080860 A080861 A080862

Keyword

nonn,easy

Author

Paul Barry, Feb 23 2003

Extensions

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

Status

approved