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A203849 a(n) = sigma_2(n)*Fibonacci(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^2*x^n/(1-x^n) = Sum_{n>=1} sigma_2(n)*x^n.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..2500

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).

EXAMPLE

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)) +...

MATHEMATICA

Table[DivisorSigma[2, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *)

PROG

(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)}

CROSSREFS

Cf. A203847, A203848, A203838, A001157 (sigma_2), A000204 (Lucas), A000045.

Sequence in context: A270631 A032340 A011854 * A271260 A270734 A271411

Adjacent sequences:  A203846 A203847 A203848 * A203850 A203851 A203852

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 12 2012

STATUS

approved

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Last modified September 30 11:32 EDT 2020. Contains 337439 sequences. (Running on oeis4.)