%I #43 May 08 2021 08:33:38
%S 2,36,7392,1688148,197412831,21085413226416,101768454084335346,
%T 60343478516053297339236,73240105330540144095414793632,
%U 1956470757376233684880813258936380492,32802418997525523144166495047229414174839,202042966989952174292936124782341088713724476716231
%N a(n) is the denominator of Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)), with Lucas(i) as defined in A000032.
%C The numerators are given in A333208.
%C See A333088 and A333089 for similar fractions for infinite sums of Fibonacci numbers.
%C Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)) is a fraction for n > 0.
%C Sum_{i >= 0} 1/Lucas(i)^2 = 1/4 + A105394, i.e., the n = 0 case, is believed to be transcendental.
%H A.H.M. Smeets, <a href="/A333209/b333209.txt">Table of n, a(n) for n = 1..54</a>
%F a(n) = denominator of (1/Fibonacci(2n)) * Sum_{i = 1..n} 1/(Lucas(2i-2)*Lucas(2i-1)).
%F Lim_{n -> inf} (A333208(n)/a(n)) / (A333208(n-1)/a(n-1)) = 1 - 1/phi = 1/phi^2 = A132318.
%F The following generalization holds: (Start)
%F Let H_(a,b) (n) be defined by H_(a,b) (0) = a, H_(a,b) (1) = b and H_(a,b) (n) = H_(a,b) (n-1) + H_(a,b) (n-2) for n > 1, then
%F Sum_{i >= 0} 1/(H_(a,b) (i)*H_(a,b) (i+2n)) = (1/Fibonacci(2n)) * Sum_{i=1..n} 1/(H_(a,b) (2i-2)*H_(a,b) (2i-1)) for n > 0, and are thus all fractions. Specially, H_(0,1) are the Fibonacci numbers A000045, H_(2,1) as here, are the Lucas numbers A000032, and H_(3,1) are the Pibonacci numbers A104449. (End)
%e These infinite sums begin: 1/2, 7/36, 551/7392, ...
%t a[n_] := Denominator[Sum[1/(LucasL[2 i - 2]*LucasL[2 i - 1]), {i, 1, n}]/Fibonacci[2 n]]; Array[a, 12] (* _Amiram Eldar_, Mar 11 2020 *)
%o (Python)
%o from math import gcd
%o f0, f1, g0, g1, snum, sden, n = 1, 1, 1, 2, 0, 1, 0
%o while n < 12:
%o n = n+1
%o snum, sden = g0*g1*snum+sden, sden*g0*g1
%o d = gcd(snum,sden*f0)
%o print(n,sden*f0//d)
%o f0, f1, g0, g1 = 2*f0+f1, f0+f1, 2*g0+g1, g0+g1 # _A.H.M. Smeets_, Nov 30 2020
%o (Python)
%o from math import gcd
%o f0, f1, g0, g1, snum, sden, n = 1, 1, 1, 2, 0, 1, 0
%o while n < 12:
%o n = n+1
%o snum, sden = g0*g1*snum+sden, sden*g0*g1
%o d = gcd(snum,sden*f0)
%o print(n,sden*f0//d)
%o f0, f1, g0, g1 = 2*f0+f1, f0+f1, 2*g0+g1, g0+g1 # _A.H.M. Smeets_, Nov 30 2020
%Y Cf. A000032, A105394, A132318, A333088, A333089, A333208.
%K nonn,frac
%O 1,1
%A _A.H.M. Smeets_, Mar 11 2020