A010008 a(0) = 1, a(n) = 18*n^2 + 2 for n>0.

1, 20, 74, 164, 290, 452, 650, 884, 1154, 1460, 1802, 2180, 2594, 3044, 3530, 4052, 4610, 5204, 5834, 6500, 7202, 7940, 8714, 9524, 10370, 11252, 12170, 13124, 14114, 15140, 16202, 17300, 18434, 19604, 20810, 22052, 23330, 24644, 25994, 27380, 28802, 30260

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The identity (18*n^2+2)^2-(9*n^2+2)*(6*n)^2=4 can be written as a(n+1)^2-A010002(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

After 20, all terms are in A000408.

Cf. A206399.

Programs

Magma

[1] cat [18*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Mathematica

Join[{1}, 18 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {20, 74, 164}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

Formula

G.f.: (1+x)*(1+16*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012

a(n) = (3*n-1)^2+(3*n+1)^2 = (n-1)^2+(n+1)^2+(4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012

E.g.f.: (x*(x+1)*18+2)*e^x-1. - Gopinath A. R., Feb 14 2012

Sum_{n>=0} 1/a(n) = 3/4+ (1/12)*Pi*coth(Pi/3) = 1.0853330948... - R. J. Mathar, May 07 2024

a(n) = 2*A247792(n), n>0. - R. J. Mathar, May 07 2024

a(n) = A069131(n)+A069131(n+1). - R. J. Mathar, May 07 2024

Extensions

More terms from Bruno Berselli, Feb 06 2012