A153387 -id:A153387 - cp's OEIS frontend

A228043 Decimal expansion of sum of reciprocals, row 5 of Wythoff array, W = A035513.

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

Offset: 0

Clark Kimberling, Aug 05 2013

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.

Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

			1/12 + 1/20 + 1/32 + ... = 0.21497141656079438829300282572973179492222...

Cf. A035513, A079586, A228040, A228041, A228042.

f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 5; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Equals A079586/4 - 5/8. - Amiram Eldar, May 22 2021

A371647 Decimal expansion of Sum_{k>=0} 1/Fibonacci(5^k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 31 2024

This constant is a transcendental number (Nyblom, 2001).

			1.20001332889036987670776409546835505643055506881380...

M. A. Nyblom, A Theorem on Transcendence of Infinite Series II, Journal of Number Theory, Vol. 91, No. 1 (2001), pp. 71-80.
Index entries for transcendental numbers.

Cf. A000045, A145232, A371648.

Similar constants: A079585, A079586, A153386, A153387, A371649.

RealDigits[Sum[1/Fibonacci[5^k], {k, 0, 10}], 10, 120][[1]]

PARI
```
suminf(k = 0, 1/fibonacci(5^k))
```

Equals Sum_{k>=0} 1/A145232(k).

A079587 Continued fraction expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k).

Original entry on oeis.org

3, 2, 1, 3, 1, 1, 13, 2, 3, 3, 2, 1, 1, 6, 3, 2, 4, 362, 2, 4, 8, 6, 30, 50, 1, 6, 3, 3, 2, 7, 2, 3, 1, 3, 2, 1, 1, 1, 1, 3, 4, 1, 1, 1, 172, 3, 26, 1, 18, 8, 2, 1, 1, 2, 8, 2, 72, 2, 3, 1, 1, 1, 1, 5, 2, 33, 1, 2, 52, 1, 3, 5, 1, 11, 1, 1, 3, 13, 2, 2, 4, 2, 14, 4, 1, 1, 10, 4, 4, 3, 5, 17, 3, 4, 2, 4, 1

Offset: 0

Benoit Cloitre, Jan 26 2003

			3.35988566624....

Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.
Wikipedia, Reciprocal Fibonacci constant

Cf. A079586 (decimal expansion), A153386, A153387.

Offset changed by Andrew Howroyd, Jul 06 2024

A357053 Decimal expansion of Sum_{k>=1} k/Fibonacci(2*k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 10 2022

This constant is transcendental (Duverney et al., 1998).

			2.39741418791652120040922449568177870852072229637554...

Daniel Duverney, Keiji Nishioka, Kumiko Nishioka, and Iekata Shiokawa, Transcendence of Jacobi's theta series and related results, in: K. Györy, et al. (eds.), Number Theory, Diophantine, Computational and Algebraic Aspects, Proceedings of the International Conference held in Eger, Hungary, July 29-August 2, 1996, de Gruyter, 1998, pp. 157-168.

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.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Index entries for transcendental numbers.

Cf. A000045, A001622, A001906, A002448, A002878, A081071, A357054.

Cf. A079586, A153386, A153387.

RealDigits[Sum[k/Fibonacci[2*k], {k, 1, 300}], 10, 100][[1]]

sumpos(k=1, k/fibonacci(2*k)) \\ Michel Marcus, Sep 10 2022

Equals Sum_{k>=1} k/A001906(k).
Equals sqrt(5) * Sum_{k>=1} 1/Lucas(2*k-1)^2 (Jennings, 1994).
Equals (1/2)*(1/phi^4 - 1)*theta_4'(1/phi^2)/theta_4(1/phi^2), where phi is the golden ratio (A001622) and theta_4 is a Jacobi theta function.

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).

A346588 Decimal expansion of the sum of reciprocals of tribonacci numbers A000213.

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Jul 24 2021

			3.7739480601970158183854024266295127497680741732225...

Cf. A000213, A079586, A084119, A093540, A153386, A153387.

RealDigits[Total[1/LinearRecurrence[{1, 1, 1}, {1, 1, 1}, 500]], 10, 105][[1]] (* Amiram Eldar, Jul 26 2021 *)

More terms from Jon E. Schoenfield, Jul 25 2021

cp's OEIS Frontend

A228043 Decimal expansion of sum of reciprocals, row 5 of Wythoff array, W = A035513.

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A371647 Decimal expansion of Sum_{k>=0} 1/Fibonacci(5^k).

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A079587 Continued fraction expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k).

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A357053 Decimal expansion of Sum_{k>=1} k/Fibonacci(2*k).

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

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A346588 Decimal expansion of the sum of reciprocals of tribonacci numbers A000213.

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