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A337668
Decimal expansion of Product_{n>=1} (1+1/Fibonacci(n)).
8
1, 3, 1, 5, 0, 9, 6, 6, 6, 5, 7, 7, 8, 4, 1, 8, 4, 3, 6, 7, 6, 1, 2, 4, 3, 3, 3, 7, 0, 6, 2, 6, 6, 5, 8, 9, 3, 2, 1, 9, 0, 1, 9, 9, 5, 4, 3, 1, 6, 4, 2, 8, 4, 7, 0, 1, 3, 5, 4, 1, 0, 0, 7, 4, 7, 1, 5, 7, 6, 9, 8, 0, 4, 6, 0, 2, 7, 6, 1, 1, 8, 0, 9, 1, 1, 8, 2, 4
OFFSET
2,2
LINKS
Daniel Duverney and Yohei Tachiya, Algebraic independence of certain infinite products involving the Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 97, No. 5 (2021), pp. 29-31; arXiv preprint, arXiv:2009.06250 [math.NT], 2020.
Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2022), Article 31; alternative link.
Eric Weisstein's World of Mathematics, Dedekind Eta Function.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Wikipedia, Theta function.
FORMULA
Equals 2 * b^(-5/4) * theta_2(b)/theta_4(b^4), where theta_i are the Jacobi theta functions and b = 1/phi = A094214 (Duverney and Tachiya, 2021). - Amiram Eldar, May 27 2021
Equals 4 * phi^(5/4) * eta(4*tau_0) / eta(tau_0), where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022). - Amiram Eldar, Mar 26 2024
EXAMPLE
13.15096665778418436761243337...
MATHEMATICA
With[{b = 1/GoldenRatio}, RealDigits[2*b^(-5/4)*EllipticTheta[2, 0, b]/EllipticTheta[4, 0, b^4], 10, 100][[1]]] (* Amiram Eldar, May 27 2021 *)
PROG
(PARI) prodinf(n=1, 1+1/fibonacci(n))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Michel Marcus, Sep 15 2020
EXTENSIONS
More terms from Jinyuan Wang, Sep 19 2020
STATUS
approved