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A205965
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a(n) = Fibonacci(n)*A001227(n) for n>=1, where A001227(n) is the number of odd divisors of n.
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4
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1, 1, 4, 3, 10, 16, 26, 21, 102, 110, 178, 288, 466, 754, 2440, 987, 3194, 7752, 8362, 13530, 43784, 35422, 57314, 92736, 225075, 242786, 785672, 635622, 1028458, 3328160, 2692538, 2178309, 14098312, 11405774, 36909860, 44791056, 48315634, 78176338, 252983944
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OFFSET
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1,3
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COMMENTS
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Compare g.f. to the Lambert series of A001227: Sum_{n>=1} x^(2*n-1)/(1 - x^(2*n-1)).
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..2500
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FORMULA
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G.f.: Sum_{n>=1} Fibonacci(2*n-1)*x^(2*n-1)/(1 - Lucas(2*n-1)*x^(2*n-1)-x^(4*n-2)).
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EXAMPLE
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G.f.: A(x) = x + x^2 + 4*x^3 + 3*x^4 + 10*x^5 + 16*x^6 + 26*x^7 + 21*x^8 +...
where A(x) = 1*1*x + 1*1*x^2 + 2*2*x^3 + 3*1*x^4 + 5*2*x^5 + 8*2*x^6 + 13*2*x^7 + 21*1*x^8 +...+ Fibonacci(n)*A001227(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1*x/(1-x-x^2) + 2*x^3/(1-4*x^3-x^6) + 5*x^5/(1-11*x^5-x^10) + 13*x^7/(1-29*x^7-x^14) + 34*x^9/(1-76*x^9-x^18) + 89*x^11/(1-199*x^11-x^22) +...
which involves odd-indexed Fibonacci and Lucas numbers.
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MATHEMATICA
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A001227[n_]:= DivisorSum[n, Mod[#, 2] &]; Table[A001227[n]*Fibonacci[n], {n, 1, 50}] (* G. C. Greubel, Jul 17 2018 *)
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PROG
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(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1, n, fibonacci(2*m-1)*x^(2*m-1)/(1-Lucas(2*m-1)*x^(2*m-1)-x^(4*m-2)+x*O(x^n))), n)}
for(n=1, 40, print1(a(n), ", "))
(PARI) a(n) = fibonacci(n)*sumdiv(n, d, d%2); \\ Michel Marcus, Jul 18 2018
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CROSSREFS
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Cf. A001227, A205964, A205966, A203847, A000204 (Lucas).
Cf. A209445 (Pell variant).
Sequence in context: A103252 A065763 A185730 * A242391 A241862 A222510
Adjacent sequences: A205962 A205963 A205964 * A205966 A205967 A205968
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Feb 03 2012
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STATUS
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approved
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