A371525 Decimal expansion of Product_{k>=1} (1 + 1/Lucas(k)).

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

Offset: 1

Author

Amiram Eldar, Mar 26 2024

Keywords

• nonn
• cons

Comments

Any two of the four constants {A337668, A337669, this, A371526} are algebraically independent over Q, while any three are not (Duverney et al., 2022).

Examples

			4.79628852318838546381037014075121584981951630809234...
		

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, A371526, A371527, A371528, A371529, A371530.

Programs

Mathematica

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

PARI

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

Formula

Equals Product_{k>=1} (1 + 1/A000032(k)).

Equals 2 * sqrt(5) * A371529.

Equals 2 * phi^(1/4) * eta(2*tau_0)^3 * eta(3*tau_0) / (eta(tau_0)^2 * 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).