A153386 -id:A153386 - cp's OEIS frontend

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

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

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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(2k).

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

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cp's OEIS Frontend

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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Extensions

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