A371527 Decimal expansion of Product_{k>=2} (1 + (-1)^k/Fibonacci(k)).

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

Offset: 1

Author

Amiram Eldar, Mar 26 2024

Keywords

• nonn
• cons

Examples

			1.13873486170719621809689508574204318763788894791573...
		

Links

• 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.
• Wikipedia, Dedekind eta function.

Crossrefs

Cf. A000045, A001622, A002390, A337668, A337669, A371525, A371526, A371528, A371529, A371530.

Programs

Mathematica

With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[2 * Sqrt[5] * Surd[GoldenRatio^5, 4] * eta[tau0]^3 * eta[4*tau0]/eta[2*tau0]^2, 10, 120][[1]]]

PARI

prodinf(k = 2, 1 + (-1)^k/fibonacci(k))

Formula

Equals Product_{k>=2} (1 + (-1)^k/A000045(k)).

Equals 6 * A337669.

Equals 2 * sqrt(5) * phi^(5/4) * 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).