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A204060 revision #6

A204060

G.f.: Sum_{n>=1} fibonacci(n^2)*x^(n^2).

1, 0, 0, 3, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 987, 0, 0, 0, 0, 0, 0, 0, 0, 75025, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14930352, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7778742049, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10610209857723, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 37889062373143906

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OFFSET

1,4

COMMENTS

Compare g.f. to the Lambert series identity: Sum_{n>=1} lamda(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^2).

Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

LINKS

Table of n, a(n) for n=1..81.

FORMULA

G.f.: Sum_{n>=1} lambda(n)*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)), where lambda(n) = A008836(n) and Lucas(n) = A000204(n).

EXAMPLE

G.f.: A(x) = x + 3*x^4 + 34*x^9 + 987*x^16 + 75025*x^25 + 14930352*x^36 +...

where A(x) = x/(1-x-x^2) + (-1)*1*x^2/(1-3*x^2+x^4) + (-1)*2*x^3/(1-4*x^3-x^6) + (+1)*3*x^4/(1-7*x^4+x^8) + (-1)*5*x^5/(1-11*x^5-x^10) + (+1)*8*x^6/(1-18*x^6+x^12) +...+ lambda(n)*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

PROG

(PARI) {a(n)=issquare(n)*fibonacci(n)}

(PARI) {lambda(n)=local(F=factor(n)); (-1)^sum(i=1, matsize(F)[1], F[i, 2])}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

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

CROSSREFS

Cf. A203847, A054783, A008836 (lambda), A000204 (Lucas), A000045.

Sequence in context: A221787 A348070 A348071 * A359780 A085393 A128980

Adjacent sequences: A204057 A204058 A204059 * A204061 A204062 A204063

KEYWORD

nonn

AUTHOR

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 12 2012

STATUS

approved