OFFSET
1,2
COMMENTS
The denominators are given in A333089.
Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)) is a fraction for n > 0.
Sum_{i > 0} 1/Fibonacci(i)^2, i.e., the n = 0 case, is known to be transcendental. See A105393.
Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n-1)) is an irrational number for n > 0. See for instance A290565 (n = 1).
LINKS
A.H.M. Smeets, Table of n, a(n) for n = 1..62
Brother Alfred Brousseau, Summation of Infinite Fibonacci Series, The Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 143-168. See (5) and (6) p. 148.
Stanley Rabinowitz, Algorithmic summation of reciprocals of products of Fibonacci numbers, The Fibonacci Quarterly, Vol. 37 (1999), pp. 122-127. See (23) and (25) p. 5.
FORMULA
a(n) = numerator of (1/Fibonacci(2n)) * Sum_{0 < i <= n} 1/(Fibonacci(2i-1)*Fibonacci(2i)).
EXAMPLE
These infinite sums begin: 1, 7/18, 143/960, ...
MATHEMATICA
a[n_] := Numerator[Sum[1/(Fibonacci[2i-1]*Fibonacci[2i]), {i, 1, n}]/Fibonacci[2n]]; Array[a, 13] (* Amiram Eldar, Mar 10 2020 *)
PROG
(PARI) a(n) = numerator(sum(i=1, n, 1/(fibonacci(2*i-1)*fibonacci(2*i)))/ fibonacci(2*n)); \\ Michel Marcus, Mar 10 2020
(Python)
from math import gcd
f0, f1, snum, sden, n = 1, 1, 0, 1, 0
while n < 13:
snum, sden, n = f0*f1*snum+sden, sden*f0*f1, n+1
d = gcd(snum, sden*f0)
print(n, snum//d)
f0, f1 = 2*f0+f1, f0+f1 # A.H.M. Smeets, May 16 2020
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
A.H.M. Smeets, Mar 07 2020
STATUS
approved