A205964 a(n) = Fibonacci(n)*A000143(n) for n>=1 with a(0)=1, where A000143(n) is the number of ways of writing n as a sum of 8 squares.

1, 16, 112, 896, 3408, 10080, 25088, 71552, 195888, 411808, 776160, 1896768, 4580352, 8194144, 14525056, 34433280, 73890768, 125562528, 219081856, 458906560, 968315040, 1686909952, 2642197824, 5579174016, 12110579712, 18907500400, 29884043168, 64236542720

Offset

0,2

Comments

Compare g.f. to the Lambert series of A000143: 1 + 16*Sum_{n>=1} n^3*x^n/(1 - (-x)^n).

Links

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

Formula

G.f.: 1 + 16*Sum_{n>=1} Fibonacci(n)*n^3*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).

Example

G.f.: A(x) = 1 + 16*x + 112*x^2 + 896*x^3 + 3408*x^4 + 10080*x^5 +...
where A(x) = 1 + 1*16*x + 1*112*x^2 + 2*448*x^3 + 3*1136*x^4 + 5*2016*x^5 + 8*3136*x^6 + 13*5504*x^7 + 21*9328*x^8 +...+ Fibonacci(n)*A000143(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 16*( 1*1*x/(1+x-x^2) + 1*8*x^2/(1-3*x^2+x^4) + 2*27*x^3/(1+4*x^3-x^6) + 3*64*x^4/(1-7*x^4+x^8) + 5*125*x^5/(1+11*x^5-x^10) + 8*216*x^6/(1-18*x^6+x^12) + 13*343*x^7/(1+29*x^7-x^14) +...).

Mathematica

Join[{1}, Table[Fibonacci[n]*SquaresR[8, n], {n, 1, 50}]] (* G. C. Greubel, Mar 05 201 *)

Prog

(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1+16*sum(m=1, n, fibonacci(m)*m^3*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. A000143, A205507, A205963, A203847, A000204 (Lucas).

Cf. A209444 (Pell variant).

Sequence in context: A035016 A000143 A258546 * A222113 A229531 A253460

Adjacent sequences: A205961 A205962 A205963 * A205965 A205966 A205967

Keyword

nonn

Author

Paul D. Hanna, Feb 03 2012

Status

approved