A204291 G.f.: Sum_{n>=1} moebius(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)), where Lucas(n) = A000204(n).

1, 0, 1, 0, 4, -3, 12, 0, 17, -10, 88, -54, 232, -28, 184, 0, 1596, -969, 4180, -1230, 4632, -198, 28656, -17388, 60020, -520, 98209, -23604, 514228, -461932, 1346268, 0, 1722688, -3570, 6672168, -5598882, 24157816, -9348, 31351552, -18606210, 165580140

Offset: 1

Author

Paul D. Hanna, Jan 14 2012

Keywords

• sign

Comments

Compare g.f. to the identity: x = Sum_{n>=1} moebius(n)*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

Examples

			G.f.: A(x) = x + x^3 + 4*x^5 - 3*x^6 + 12*x^7 + 17*x^9 - 10*x^10 + 88*x^11 +...
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^5/(1-11*x^5-x^10) + x^6/(1-18*x^6+x^12) +...+ moebius(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1024

Crossrefs

Cf. A203847, A000204 (Lucas), A000045, A008683.

Programs

Maple

with(numtheory): seq(add(mobius(d)*combinat[fibonacci](n)/combinat[fibonacci](d), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 09 2025

PARI

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

Formula

a(k) = 0 iff k = 2^n for n>=1.

a(n) = Fibonacci(n) * Sum_{d|n} mu(d)/Fibonacci(d). - Ridouane Oudra, Apr 09 2025