A085473 a(n) = 6*n^2 + 3*n + 1.

1, 10, 31, 64, 109, 166, 235, 316, 409, 514, 631, 760, 901, 1054, 1219, 1396, 1585, 1786, 1999, 2224, 2461, 2710, 2971, 3244, 3529, 3826, 4135, 4456, 4789, 5134, 5491, 5860, 6241, 6634, 7039, 7456, 7885, 8326, 8779, 9244, 9721, 10210, 10711, 11224, 11749, 12286, 12835

Offset: 0

Author

Paul Barry, Jul 01 2003

Keywords

• nonn
• easy

Comments

T(n,3) of A085475.

Sequence found by reading the line from 1, in the direction 1, 10,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 09 2011

Sums of the triangular numbers from A000217(2*n-1) to A000217(2*n+1), with A000217(-1) = 0. - Bruno Berselli, Sep 04 2018

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

Cf. A000217, A001318, A016813, A085474, A085475.

Programs

Mathematica

Table[3 n (2 n + 1) + 1, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
Table[Binomial[2 n + 3, 3] - Binomial[2 n, 3], {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 10, 31}, 50] (* Harvey P. Dale, Nov 15 2011 *)

PARI

x='x+O('x^50); Vec((1+7*x+4*x^2)/(1-x)^3) \\ G. C. Greubel, Jun 13 2017

PARI

for(n=0,25, print1(6*n^2 + 3*n + 1, ", ")) \\ G. C. Greubel, Jun 13 2017

Formula

G.f.: (1 + 7*x + 4*x^2)/(1 - x)^3.

a(n) = binomial(2*n+3,3) - binomial(2*n,3).

a(n) = 12*n + a(n-1) - 3 for n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

a(0)=1, a(1)=10, a(2)=31; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Nov 15 2011

E.g.f.: exp(x)*(1 + 9*x + 6*x^2). - Elmo R. Oliveira, Oct 18 2024