A010012 a(0) = 1, a(n) = 22*n^2 + 2 for n>0.

1, 24, 90, 200, 354, 552, 794, 1080, 1410, 1784, 2202, 2664, 3170, 3720, 4314, 4952, 5634, 6360, 7130, 7944, 8802, 9704, 10650, 11640, 12674, 13752, 14874, 16040, 17250, 18504, 19802, 21144, 22530, 23960, 25434, 26952, 28514, 30120, 31770, 33464, 35202, 36984

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Bruno Berselli, Feb 06 2012: (Start)

First trisection of A008259.

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=2, s=3. After 1, all terms are in A000408. (End)

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

Mathematica

Join[{1}, 22 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1},LinearRecurrence[{3,-3,1},{24,90,200},50]] (* Harvey P. Dale, Jul 20 2013 *)
CoefficientList[Series[(1 + x) (1 + 20 x + x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 03 2015 *)

Formula

G.f.: (1+x)*(1+20*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012

E.g.f.: (x*(x+1)*22+2)*e^x-1. - Gopinath A. R., Feb 14 2012

Sum_{n>=0} 1/a(n) = 3/4 + sqrt(11)/44*Pi*coth( Pi/sqrt(11)) = 1.0706480516966... - R. J. Mathar, May 07 2024

a(n) = A069173(n)+A069173(n+1). - R. J. Mathar, May 07 2024