A203802 a(n) = Fibonacci(n) * Sum_{d|n} -(-1)^(n/d) / Fibonacci(d).

1, 0, 3, -5, 6, -3, 14, -48, 52, -10, 90, -329, 234, -28, 1038, -2349, 1598, -1044, 4182, -12750, 17262, -198, 28658, -135285, 90031, -520, 300405, -554974, 514230, -464658, 1346270, -5188656, 5326470, -3570, 11782764, -34556612, 24157818, -9348, 95140422, -256249218, 165580142

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{n>=1} x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).

Example

G.f.: A(x) = x + 3*x^3 - 5*x^4 + 6*x^5 - 3*x^6 + 14*x^7 - 48*x^8 + 52*x^9 +...
where A(x) = x/(1+x-x^2) + x^2/(1+3*x^2+x^4) + x^3/(1+4*x^3-x^6) + x^4/(1+7*x^4+x^8) + x^5/(1+11*x^5-x^10) + x^6/(1+18*x^6+x^12) +...+ x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
Illustration of terms.
a(1) = 1; a(2) = 1*(-1/1 + 1/1) = 0; a(3) = 2*(1/1 + 1/2) = 3;
a(4) = 3*(-1/1 - 1/1 + 1/3) = -5; a(5) = 5*(1/1 + 1/5) = 6;
a(6) = 8*(-1/1 + 1/1 - 1/2 + 1/8) = -3; a(7) = 13*(1/1 + 1/13) = 14;
a(8) = 21*(-1/1 - 1/1 - 1/3 + 1/21) = -48; ...

Prog

(PARI) {a(n)=fibonacci(n) * sumdiv(n, d, -(-1)^(n/d) / fibonacci(d))}
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1, n, x^m/(1 + Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}

Crossrefs

Cf. A111075, A000045, A000204 (Lucas).

Sequence in context: A152713 A218802 A236101 * A306554 A078064 A091517

Adjacent sequences: A203799 A203800 A203801 * A203803 A203804 A203805

Keyword

sign

Author

Paul D. Hanna, Jan 11 2012

Status

approved