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 A333089 a(n) is the denominator of Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)). 4
 1, 18, 960, 76440, 37437400, 157024707840, 2777798704721040, 4169982785629476816, 331259342780844858796416, 743322803326470921519628462800, 163037651356772148158514292729628880, 187555796967791569325602741834073910082560, 3838658658324493911932517275499048601188128008800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The numerators are given in A333088. 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) = denominator 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_] := Denominator[Sum[1/(Fibonacci[2i-1]*Fibonacci[2i]), {i, 1, n}] / Fibonacci[2n]]; Array[a, 13] (* Amiram Eldar, Mar 10 2020 *) PROG (PARI) a(n) = denominator(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, sden*f0//d)     f0, f1 = 2*f0+f1, f0+f1 # A.H.M. Smeets, May 16 2020 CROSSREFS Cf. A105393, A290565, A333088 (numerator). Sequence in context: A242446 A160013 A123786 * A129009 A042409 A073960 Adjacent sequences:  A333086 A333087 A333088 * A333090 A333091 A333092 KEYWORD nonn,frac AUTHOR A.H.M. Smeets, Mar 07 2020 STATUS approved

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Last modified June 24 18:46 EDT 2021. Contains 345419 sequences. (Running on oeis4.)