A004538 a(n) = 3*n^2 + 3*n - 1.

-1, 5, 17, 35, 59, 89, 125, 167, 215, 269, 329, 395, 467, 545, 629, 719, 815, 917, 1025, 1139, 1259, 1385, 1517, 1655, 1799, 1949, 2105, 2267, 2435, 2609, 2789, 2975, 3167, 3365, 3569, 3779, 3995, 4217, 4445

Offset: 0

Author

N. J. A. Sloane, Eric T. Lane (ERICLANE(AT)UTCVM.UTC.EDU)

Keywords

• sign
• easy

Comments

Numbers k such that (4*k + 7)/3 is a square. - Bruno Berselli, Sep 11 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

First differences of A033445.

Cf. A028896, A003215, A077588, A000538, A000330.

Programs

Magma

[3*n^2 + 3*n -1: n in [0..50]]; // G. C. Greubel, Sep 10 2018

Mathematica

Table[5*Sum[k^4,{k,1,n}]/Sum[k^2,{k,1,n}], {n,1,20}] (* Alexander Adamchuk, Apr 12 2006 *)
Table[3n^2+3n-1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{-1,5,17},40] (* Harvey P. Dale, Jan 18 2019 *)

PARI

a(n)=3*n^2+3*n-1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

From Alexander Adamchuk, Apr 12 2006: (Start)

a(n) = 5 * Sum_{k=1..n} k^4 / Sum_{k=1..n} k^2, n > 0.

a(n) = 5 * A000538(n) / A000330(n), n > 0. (End)

a(n) = a(n-1) + 6*n with a(0)=-1. - Vincenzo Librandi, Nov 18 2010

From G. C. Greubel, Sep 10 2018: (Start)

G.f.: (-1 + 8*x - x^2)/(1 - x)^3.

E.g.f.: (-1 + 6*x + 3*x^2)*exp(x). (End)

Sum_{n>=0} 1/a(n) = ( psi(1/2+sqrt(21)/6) - psi(1/2-sqrt(21)/6)) /sqrt(21) = -0.6286929... R. J. Mathar, Apr 24 2024