A203849 a(n) = sigma_2(n)*Fibonacci(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.
1, 5, 20, 63, 130, 400, 650, 1785, 3094, 7150, 10858, 30240, 39610, 94250, 158600, 336567, 463130, 1175720, 1513522, 3693690, 5473000, 10803710, 15188210, 39412800, 48841275, 103184050, 161062760, 333701550, 432980818, 1081652000, 1295110778, 2973391785, 4299985160
Offset: 1
Keywords
Examples
G.f.: A(x) = x + 5*x^2 + 20*x^3 + 63*x^4 + 130*x^5 + 400*x^6 + 650*x^7 +... where A(x) = x/(1-x-x^2) + 2^2*1*x^2/(1-3*x^2+x^4) + 3^2*2*x^3/(1-4*x^3-x^6) + 4^2*3*x^4/(1-7*x^4+x^8) + 5^2*5*x^5/(1-11*x^5-x^10) + 6^2*8*x^6/(1-18*x^6+x^12) +...+ n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..2500
Programs
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Mathematica
Table[DivisorSigma[2, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *) -
PARI
{a(n)=sigma(n,2)*fibonacci(n)} -
PARI
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)} {a(n)=polcoeff(sum(m=1,n,m^2*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
Formula
G.f.: Sum_{n>=1} n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_2(n)*fibonacci(n)*x^n, where Lucas(n) = A000204(n).
Comments