A337669 Decimal expansion of Product_{n>=3} (1-1/Fibonacci(n)).
1, 8, 9, 7, 8, 9, 1, 4, 3, 6, 1, 7, 8, 6, 6, 0, 3, 6, 3, 4, 9, 4, 8, 2, 5, 1, 4, 2, 9, 0, 3, 4, 0, 5, 3, 1, 2, 7, 2, 9, 8, 1, 4, 9, 1, 3, 1, 9, 2, 8, 8, 7, 5, 2, 2, 9, 0, 6, 8, 9, 0, 7, 0, 7, 1, 0, 8, 2, 0, 4, 6, 9, 4, 5, 7, 0, 3, 7, 0, 4, 4, 5, 6, 6, 7, 9, 7
Offset: 0
Examples
0.18978914361786603634948251429...
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, Dedekind eta function.
- Wikipedia, Theta function.
Programs
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Mathematica
With[{b = 1/GoldenRatio}, RealDigits[(Sqrt[5]/6)*b^(-5/4) * EllipticTheta[2, 0, b] * EllipticTheta[3, 0, b] * EllipticTheta[4, 0, b]/EllipticTheta[4, 0, b^4], 10, 100][[1]]] (* Amiram Eldar, May 27 2021 *) -
PARI
prodinf(n=3, 1-1/fibonacci(n))
Formula
Equals (sqrt(5)/6) * b^(-5/4) * theta_2(b) * theta_3(b) * theta_4(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 (sqrt(5) * phi^(5/4) / 3) * eta(tau_0)^3 * eta(4*tau_0) / eta(2*tau_0)^2, 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
Extensions
More terms from Jinyuan Wang, Sep 19 2020