A158933 -id:A158933 - cp's OEIS frontend

A324007 Decimal expansion of the sum of reciprocals of the products of 3 consecutive Fibonacci numbers.

7, 1, 0, 8, 5, 5, 3, 5, 1, 4, 2, 9, 3, 2, 8, 4, 1, 6, 8, 8, 7, 6, 9, 4, 4, 9, 0, 3, 8, 4, 2, 7, 0, 8, 3, 3, 0, 4, 5, 1, 1, 8, 0, 4, 8, 4, 1, 0, 3, 0, 8, 6, 3, 9, 9, 7, 4, 9, 7, 3, 5, 1, 4, 9, 3, 6, 9, 6, 4, 2, 3, 8, 2, 6, 1, 1, 3, 5, 4, 4, 8, 4, 1, 7, 5, 8, 8, 4, 1, 6, 8, 1, 7, 1, 4, 8, 5, 8, 5, 7, 6, 8, 5, 4, 9

Offset: 0

Robert G. Wilson v, Feb 11 2019

			0.71085535142932841688769449038427083304511804841030863997497351493696423826...

Brother Alfred Brousseau, Summation of Infinite Fibonacci Series, The Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 143-168.
R. S. Melham, On Some Reciprocal Sums of Brousseau: An Alternative Approach to That of Carlitz, Fibonacci Quarterly, Vol. 41, No. 1 (2003), pp. 59-62.

Cf. A000045, A065563, A079586, A158933, A290565, A322711, A324008.

RealDigits[ Sum[ N[ 1/Product[ Fibonacci@j, {j, k, k + 2}], 128], {k, 177}], 10, 111][[1]]

suminf(n=1, 1/(fibonacci(n)*fibonacci(n+1)*fibonacci(n+2))) \\ Michel Marcus, Feb 19 2019

From Amiram Eldar, Feb 09 2023: (Start)
Equals Sum_{k>=1} 1/A065563(k).
Equals 1 - A158933 (Melham, 2003). (End)

A201615 Decimal expansion of Sum_{n>=1} 1/F(n)^n, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

2, 1, 3, 7, 6, 6, 9, 5, 0, 9, 6, 7, 2, 6, 9, 8, 4, 3, 3, 3, 1, 7, 1, 4, 9, 8, 1, 6, 9, 0, 3, 2, 6, 1, 9, 4, 1, 9, 0, 3, 9, 6, 6, 6, 3, 1, 7, 4, 4, 2, 0, 9, 7, 5, 8, 4, 7, 2, 1, 2, 1, 4, 7, 1, 0, 5, 2, 3, 8, 7, 1, 0, 1, 1, 6, 3, 4, 5, 5, 0, 5, 2, 5, 3, 9, 6, 5, 8, 8, 6, 2, 6, 3, 0, 5, 3, 3, 3, 6, 6, 0, 8, 6, 8, 0

Offset: 1

Michel Lagneau, Dec 03 2011

			2.13766950967269843331714981... = 1/1^1 + 1/1^2+ 1/2^3+ 1/3^4 +1/5^5 +1/8^6 +...

Cf. A079586, A158933, A000045.

with(combinat,fibonacci):Digits:=120:s:=sum( evalf(1/ fibonacci(n)^n),n=1..200):print(s):

digits = 105; NSum[1/Fibonacci[n]^n, {n, 1, Infinity}, NSumTerms -> digits, WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)

suminf(n=1, 1/fibonacci(n)^n); \\ Michel Marcus, Feb 21 2014

A201616 Decimal expansion of Sum_{n = 1 .. infinity} (-1)^(n+1)/F(n)^n where F=A000045 is the Fibonacci sequence.

Original entry on oeis.org

1, 1, 2, 9, 7, 0, 5, 2, 2, 2, 0, 0, 5, 9, 7, 7, 4, 2, 2, 3, 8, 0, 4, 0, 6, 7, 7, 9, 0, 4, 2, 8, 7, 9, 4, 3, 4, 0, 8, 6, 1, 9, 1, 4, 5, 0, 2, 3, 1, 6, 4, 4, 8, 6, 2, 1, 1, 2, 1, 0, 5, 0, 7, 6, 7, 7, 7, 0, 1, 9, 5, 3, 8, 3, 2, 7, 3, 0, 7, 9, 8, 9, 2, 9, 2, 6, 3, 4, 6, 4, 8, 2, 2, 8, 9, 4, 3, 8, 9, 6, 9, 3, 7, 8, 8

Offset: 0

Michel Lagneau, Dec 03 2011

			0.1129705222005977422380406779... = 1/1^1 - 1/1^2 + 1/2^3 - 1/3^4 + 1/5^5 - ...

Cf. A079586, A158933, A000045, A201615.

with(combinat,fibonacci):Digits:=120:s:=sum( evalf(((-1)^(n+1))/ fibonacci(n)^n),n=1..200):print(s):

RealDigits[N[Sum[((-1)^(n+1))/Fibonacci[n]^n,{n,1,105}],105]][[1]]

-suminf(n=1,(-1)^n/fibonacci(n)^n) \\ Charles R Greathouse IV, Dec 05 2011

A338612 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/L(k) where L(k) is the k-th Lucas number (A000032).

Original entry on oeis.org

8, 3, 0, 5, 0, 2, 8, 2, 1, 5, 8, 6, 8, 7, 6, 6, 8, 2, 3, 1, 6, 9, 3, 6, 4, 8, 6, 2, 5, 1, 0, 5, 9, 5, 1, 9, 1, 7, 7, 3, 0, 4, 6, 2, 1, 4, 3, 0, 4, 0, 8, 2, 8, 0, 1, 4, 6, 0, 2, 6, 4, 1, 3, 9, 0, 7, 9, 1, 0, 4, 9, 8, 4, 8, 6, 0, 4, 3, 0, 0, 6, 7, 4, 9, 3, 3, 0

Offset: 0

Amiram Eldar, Nov 03 2020

André-Jeannin (1989) proved that this constant is irrational, and Tachiya (2004) proved that it does not belong to the quadratic number field Q(sqrt(5)).

			0.83050282158687668231693648625105951917730462143040...

Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541.
Yohei Tachiya, Irrationality of certain Lambert series, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
Eric Weisstein's World of Mathematics, Reciprocal Lucas Constant.

Cf. A000032, A000045, A001622, A079586, A093540, A153415, A153416, A158933.

RealDigits[Sum[(-1)^(n+1)/LucasL[n], {n, 1, 1000}], 10, 120][[1]]

Equals A153416 - A153415.
Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/Fibonacci(2*k).
Equals Sum_{k>=1} (-1)^(k+1)/(phi^k + (1-phi)^k), where phi is the golden ratio (A001622).
Equals Sum_{k>=0} 1/(phi^(2*k+1) + (-1)^k).

A357054 Decimal expansion of Sum_{k>=1} (-1)^(k+1)k/Fibonacci(2k).

Original entry on oeis.org

5, 8, 0, 0, 0, 4, 7, 3, 9, 5, 0, 7, 7, 7, 0, 6, 8, 0, 0, 6, 7, 4, 7, 0, 9, 8, 1, 8, 9, 5, 5, 2, 2, 8, 0, 2, 6, 9, 8, 5, 0, 1, 2, 6, 0, 9, 6, 4, 6, 1, 6, 3, 9, 0, 1, 5, 7, 7, 5, 6, 1, 0, 0, 1, 7, 7, 6, 7, 3, 7, 5, 7, 5, 2, 1, 9, 9, 7, 8, 4, 8, 9, 4, 9, 2, 1, 0, 4, 4, 7, 8, 6, 6, 9, 4, 0, 2, 2, 3, 7, 1, 4, 1, 1, 5

Offset: 0

Amiram Eldar, Sep 10 2022

			0.58000473950777068006747098189552280269850126096461...

Daniel Duverney and Iekata Shiokawa, On series involving Fibonacci and Lucas numbers I, AIP Conference Proceedings, Vol. 976, No. 1. American Institute of Physics, 2008, pp. 62-76.
Derek Jennings, On reciprocals of Fibonacci and Lucas numbers, Fibonacci Quarterly, Vol. 32, No. 1 (1994), pp. 18-21.

Cf. A000045, A001519, A001906, A081068, A357053.

Cf. A079586, A153386, A153387, A158933, A265288.

RealDigits[Sum[(-1)^(k+1)*k/Fibonacci[2*k], {k, 1, 300}], 10, 100][[1]]

sumalt(k=1, (-1)^(k+1)*k/fibonacci(2*k)) \\ Michel Marcus, Sep 10 2022

Equals Sum_{k>=1} (-1)^(k+1)*k/A001906(k).

Equals (1/sqrt(5)) * Sum_{k>=1} 1/Fibonacci(2*k-1)^2 (Jennings, 1994).

cp's OEIS Frontend

A324007 Decimal expansion of the sum of reciprocals of the products of 3 consecutive Fibonacci numbers.

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A201615 Decimal expansion of Sum_{n>=1} 1/F(n)^n, where F=A000045 (Fibonacci numbers).

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A201616 Decimal expansion of Sum_{n = 1 .. infinity} (-1)^(n+1)/F(n)^n where F=A000045 is the Fibonacci sequence.

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A338612 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/L(k) where L(k) is the k-th Lucas number (A000032).

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A357054 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*k/Fibonacci(2*k).

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A357054 Decimal expansion of Sum_{k>=1} (-1)^(k+1)k/Fibonacci(2k).