A010002 a(0) = 1, a(n) = 9*n^2 + 2 for n>0.

1, 11, 38, 83, 146, 227, 326, 443, 578, 731, 902, 1091, 1298, 1523, 1766, 2027, 2306, 2603, 2918, 3251, 3602, 3971, 4358, 4763, 5186, 5627, 6086, 6563, 7058, 7571, 8102, 8651, 9218, 9803, 10406, 11027, 11666, 12323, 12998, 13691, 14402, 15131, 15878, 16643

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=1, s=2. After 1, all terms are in A000408. [Bruno Berselli, Feb 06 2012]

The identity (18*n^2+2)^2-(9*n^2+2)*(6*n)^2 = 4 can be written as A010008(n+1)^2-a(n+1)*A008588(n+1)^2 = 4. - Vincenzo Librandi, Feb 07 2012

Links

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

Crossrefs

Cf. A206399.

Programs

Magma

[1] cat [9*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Mathematica

Join[{1}, 9 Range[43]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {11, 38, 83}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

PARI

A010002(n)=9*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

Formula

G.f.: (1+x)*(1+7*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012

E.g.f.: (x*(x+1)*9+2)*e^x-1. - Gopinath A. R., Feb 14 2012

Sum_{n>=0} 1/a(n) = 3/4+sqrt(2)/12 *Pi*coth(Pi/3*sqrt 2) = 1.1606262038.. - R. J. Mathar, May 07 2024

Extensions

More terms from Bruno Berselli, Feb 06 2012