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A225528
a(n) = sigma(n)*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.
3
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
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
Sequence in context: A187087 A072861 A183371 * A039785 A303692 A119575
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 09 2013
STATUS
approved