A225528 a(n) = sigma(n)*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.

1, 9, 16, 49, 66, 216, 232, 705, 988, 2214, 2388, 9016, 7294, 20232, 32736, 68417, 64278, 225342, 186980, 635334, 783232, 1425708, 1537896, 6220920, 5200591, 11400606, 17568160, 39796232, 34495530, 133955856, 96331168, 306863361, 378297408, 688610322, 990395472, 3038060662

Offset: 1

Author

Paul D. Hanna, May 09 2013

Keywords

• nonn

Examples

			L.g.f.: L(x) = x + 9*x^2/2 + 16*x^3/3 + 49*x^4/4 + 66*x^5/5 + 216*x^6/6 +...
which is equivalent to:
L(x) = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 + 8*29*x^7/7 + 15*47*x^8/8 +...+ sigma(n)*Lucas(n)*x^n/n +...
where exponentiation yields the g.f. of A156234:
exp(L(x)) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 63*x^5 + 170*x^6 + 355*x^7 +...+ A156234(n)*x^n +...
and equals the product:
exp(L(x)) = 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6) * (1-7*x^4+x^8) * (1-11*x^5-x^10) * (1-18*x^6+x^12) *...* (1 - Lucas(n)*x^n + (-x^2)^n) *...).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A156234, A225525.

Programs

PARI

{a(n)=sigma(n)*(fibonacci(n-1)+fibonacci(n+1))}
for(n=1,40,print1(a(n),", "))

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(sum(m=1, n, -log(1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=1,40,print1(a(n),", "))

Formula

L.g.f.: Sum_{n>=1} -log(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} a(n)*x^n/n.

Logarithmic derivative of A156234.