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A322711
Decimal expansion of the sum of reciprocals of the products of 9 consecutive Fibonacci numbers.
3
4, 5, 7, 1, 5, 2, 2, 7, 6, 2, 0, 6, 4, 8, 1, 8, 3, 7, 2, 5, 9, 8, 4, 4, 5, 5, 7, 2, 8, 8, 9, 5, 1, 8, 5, 4, 9, 1, 1, 3, 7, 2, 6, 0, 1, 2, 5, 5, 7, 9, 3, 8, 1, 5, 8, 9, 6, 0, 7, 5, 1, 7, 8, 7, 0, 5, 4, 0, 1, 1, 3, 3, 3, 7, 6, 6, 7, 8, 6, 3, 4, 2, 1, 2, 1, 9, 5
OFFSET
-6,1
LINKS
Brother Alfred Brousseau, Summation of Infinite Fibonacci Series, The Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 143-168. See (23) p. 167.
Stanley Rabinowitz, Algorithmic summation of reciprocals of products of Fibonacci numbers, The Fibonacci Quarterly, Vol. 37 (1999), pp. 122-127, alternative link. See (29) p. 127 or p. 7.
FORMULA
Equals to (319/16380) * (Sum_{k>=1} 1/F(k) - 46816051/13933920), where F(k) is the k-th Fibonacci number.
EXAMPLE
4.57152276206481837259844557288951854911372601255793... * 10^(-7).
MATHEMATICA
digits = 100; f[n_] := Product[Fibonacci[k], {k, n, n+8}]; NSum[1/f[n], {n, 1, Infinity}, WorkingPrecision -> digits, NSumTerms -> digits] // RealDigits[#, 10, digits] & // First (* after Jean-François Alcover at A079586 *)
RealDigits[ Sum[ N[ 1/Product[ Fibonacci@j, {j, k, k + 8}], 128], {k, 59}], 10, 111][[1]] (* Robert G. Wilson v, Feb 11 2019 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Dec 24 2018
STATUS
approved