A371529 - OEIS

%I #4 Mar 27 2024 21:30:13

%S 1,0,7,2,4,8,2,7,1,7,7,5,5,1,3,0,6,2,5,8,8,5,3,7,8,8,1,6,5,2,6,6,0,8,

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

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

%N Decimal expansion of Product_{k>=2} (1 + (-1)^k/Lucas(k)).

%H Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, <a href="https://doi.org/10.1007/s40993-022-00328-7">A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers</a>, Research in Number Theory, Vol. 8 (2022), Article 31; <a href="https://www2.math.kyushu-u.ac.jp/~mkaneko/papers/DEKT.pdf">alternative link</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dedekind_eta_function">Dedekind eta function</a>.

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

%F Equals A371525 / (2*sqrt(5)).

%F Equals (phi^(1/4) / sqrt(5)) * 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).

%e 1.07248271775513062588537881652660869304392049333099...

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

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

%Y Cf. A000032, A001622, A002390, A337668, A337669, A371525, A371526, A371527, A371528, A371530.

%K nonn,cons

%O 1,3

%A _Amiram Eldar_, Mar 26 2024