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%I #54 May 08 2021 08:33:24
%S 1,7,143,4351,814001,1304114687,8811986820779,5052800260335941,
%T 153317149364862950801,131408899191108437793754033,
%U 11009306212815764937387730291387,4837569887867603346019952058036959933,37818210546715267110622871226615561517197713
%N a(n) is the numerator of Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)).
%C The denominators are given in A333089.
%C Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)) is a fraction for n > 0.
%C Sum_{i > 0} 1/Fibonacci(i)^2, i.e., the n = 0 case, is known to be transcendental. See A105393.
%C Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n-1)) is an irrational number for n > 0. See for instance A290565 (n = 1).
%H A.H.M. Smeets, <a href="/A333088/b333088.txt">Table of n, a(n) for n = 1..62</a>
%H Brother Alfred Brousseau, <a href="https://www.mathstat.dal.ca/FQ/Scanned/7-2/brousseau1.pdf">Summation of Infinite Fibonacci Series</a>, The Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 143-168. See (5) and (6) p. 148.
%H Stanley Rabinowitz, <a href="https://www.mathstat.dal.ca/FQ/Scanned/37-2/rabinowitz1.pdf">Algorithmic summation of reciprocals of products of Fibonacci numbers</a>, The Fibonacci Quarterly, Vol. 37 (1999), pp. 122-127. See (23) and (25) p. 5.
%F a(n) = numerator of (1/Fibonacci(2n)) * Sum_{0 < i <= n} 1/(Fibonacci(2i-1)*Fibonacci(2i)).
%e These infinite sums begin: 1, 7/18, 143/960, ...
%t a[n_] := Numerator[Sum[1/(Fibonacci[2i-1]*Fibonacci[2i]),{i,1,n}]/Fibonacci[2n]]; Array[a, 13] (* _Amiram Eldar_, Mar 10 2020 *)
%o (PARI) a(n) = numerator(sum(i=1, n, 1/(fibonacci(2*i-1)*fibonacci(2*i)))/ fibonacci(2*n)); \\ _Michel Marcus_, Mar 10 2020
%o (Python)
%o from math import gcd
%o f0, f1, snum, sden, n = 1, 1, 0, 1, 0
%o while n < 13:
%o snum, sden, n = f0*f1*snum+sden, sden*f0*f1, n+1
%o d = gcd(snum,sden*f0)
%o print(n,snum//d)
%o f0, f1 = 2*f0+f1, f0+f1 # _A.H.M. Smeets_, May 16 2020
%Y Cf. A105393, A290565, A333089 (denominator).
%K nonn,frac
%O 1,2
%A _A.H.M. Smeets_, Mar 07 2020
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