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

1,2

Links

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

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.

Example

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) *...).

Prog

(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), ", "))

Crossrefs

Cf. A156234, A225525.

Sequence in context: A187087 A072861 A183371 * A039785 A303692 A119575

Adjacent sequences: A225525 A225526 A225527 * A225529 A225530 A225531

Keyword

nonn

Author

Paul D. Hanna, May 09 2013

Status

approved