A205963 a(n) = Fibonacci(n)*A000118(n) for n>=1 with a(0)=1, where A000118(n) is the number of ways of writing n as a sum of 4 squares.

1, 8, 24, 64, 72, 240, 768, 832, 504, 3536, 7920, 8544, 13824, 26096, 72384, 117120, 23688, 229968, 806208, 668960, 974160, 2802176, 5100768, 5502144, 4451328, 18606200, 40788048, 62853760, 61019712, 123414960, 479255040, 344644864, 52279416, 1353437952, 2463647184

Offset

0,2

Comments

Compare g.f. to the Lambert series of A000118: 1 + 8*Sum_{n>=1} n*x^n/(1 + (-x)^n).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1 + 8*Sum_{n>=1} Fibonacci(n)*n*x^n/(1 + Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).

Example

G.f.: A(x) = 1 + 8*x + 24*x^2 + 64*x^3 + 72*x^4 + 240*x^5 + 768*x^6 +...
where A(x) = 1 + 1*8*x + 1*24*x^2 + 2*32*x^3 + 3*24*x^4 + 5*48*x^5 + 8*96*x^6 + 13*64*x^7 + 21*24*x^8 +...+ Fibonacci(n)*A000118(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 8*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1+3*x^2+x^4) + 2*3*x^3/(1-4*x^3-x^6) + 3*4*x^4/(1+7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 8*6*x^6/(1+18*x^6+x^12) + 13*7*x^7/(1-29*x^7-x^14) +...).

Mathematica

Join[{1}, Table[Fibonacci[n]*SquaresR[4, n], {n, 1, 50}]] (* G. C. Greubel, Mar 09 2017 *)

Prog

(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1+8*sum(m=1, n, fibonacci(m)*m*x^m/(1+Lucas(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
for(n=0, 31, print1(a(n), ", "))

Crossrefs

Cf. A000118, A205507, A205964, A203847, A000204 (Lucas).

Cf. A209443 (Pell variant).

Sequence in context: A060602 A066605 A066497 * A208895 A111071 A090336

Adjacent sequences: A205960 A205961 A205962 * A205964 A205965 A205966

Keyword

nonn

Author

Paul D. Hanna, Feb 03 2012

Status

approved