A322711 Decimal expansion of the sum of reciprocals of the products of 9 consecutive Fibonacci numbers.

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

Robert G. Wilson v, Table of n, a(n) for n = -6..10000

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

Cf. A000045, A079586, A290565.

Sequence in context: A112247 A319260 A237196 * A057055 A177883 A245422

Adjacent sequences: A322708 A322709 A322710 * A322712 A322713 A322714

Keyword

nonn,cons

Author

Amiram Eldar, Dec 24 2018

Status

approved