A371530 Decimal expansion of Product_{k>=1} (1 - (-1)^k/Lucas(k)).

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

Offset: 1

Author

Amiram Eldar, Mar 26 2024

Keywords

• nonn
• cons

Examples

			1.50218004433245676912076258176556998802715258088888...
		

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. A000032, A001622, A002390, A337668, A337669, A371525, A371526, A371527, A371528, A371529.

Programs

Mathematica

With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[2 * Surd[GoldenRatio, 4] * eta[2*tau0]^2 * eta[6*tau0]/(eta[3*tau0] * eta[4*tau0]), 10, 120][[1]]]

PARI

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

Formula

Equals Product_{k>=2} (1 - (-1)^k/A000032(k)).

Equals (2*sqrt(5)) * A371526.

Equals 2 * phi^(1/4) * eta(2*tau_0)^2 * eta(6*tau_0) / (eta(3*tau_0) * eta(4*tau_0)), where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022).