A300895 L.g.f.: log(Product_{k>=2} (1 + x^Fibonacci(k))) = Sum_{n>=1} a(n)*x^n/n.

1, 1, 4, -3, 6, -2, 1, 5, 4, -4, 1, -6, 14, 1, 9, -11, 1, -2, 1, -8, 25, 1, 1, 2, 6, -12, 4, -3, 1, -7, 1, -11, 4, 35, 6, -6, 1, 1, 17, 0, 1, -23, 1, -3, 9, 1, 1, -14, 1, -4, 4, -16, 1, -2, 61, 5, 4, 1, 1, -11, 1, 1, 25, -11, 19, -2, 1, -37, 4, -4, 1, 2, 1, 1, 9, -3, 1, -15, 1, -16, 4, 1, 1, -27, 6

Offset

1,3

Links

Table of n, a(n) for n=1..85.

Formula

G.f.: Sum_{k>=2} Fibonacci(k)*x^Fibonacci(k)/(1 + x^Fibonacci(k)).

a(n) = n + 1 if n is an odd prime Fibonacci number (A005478 except a(1) = 2).

Example

L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 - 3*x^4/4 + 6*x^5/5 - 2*x^6/6 + x^7/7 + 5*x^8/8 + 4*x^9/9 - 4*x^10/10 + ...
exp(L(x)) = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^7 + 3*x^8 + 2*x^9 + 2*x^10 + ... + A000119(n)*x^n + ...

Mathematica

nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^Fibonacci[k]), {k, 2, 14}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 85; Rest[CoefficientList[Series[Sum[Fibonacci[k] x^Fibonacci[k]/(1 + x^Fibonacci[k]), {k, 2, 14}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[(5 #^2 + 4)^(1/2)] || IntegerQ[(5 #^2 - 4)^(1/2)] &], {n, 85}]

Crossrefs

Cf. A000045, A000119, A005092, A005478.

Sequence in context: A319614 A010653 A153200 * A317698 A179103 A045814

Adjacent sequences: A300892 A300893 A300894 * A300896 A300897 A300898

Keyword

sign

Author

Ilya Gutkovskiy, Mar 14 2018

Status

approved