A205965 a(n) = Fibonacci(n)*A001227(n) for n>=1, where A001227(n) is the number of odd divisors of n.
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
Offset: 1
Keywords
Examples
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.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..2500
Programs
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Mathematica
A001227[n_]:= DivisorSum[n, Mod[#, 2] &]; Table[A001227[n]*Fibonacci[n], {n, 1, 50}] (* G. C. Greubel, Jul 17 2018 *)
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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
Formula
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)).
Comments