login
A371526
Decimal expansion of Product_{k>=2} (1 - 1/Lucas(k)).
6
3, 3, 5, 8, 9, 7, 6, 6, 9, 3, 5, 7, 1, 0, 2, 1, 0, 3, 1, 4, 7, 6, 6, 5, 7, 2, 6, 6, 3, 1, 2, 2, 6, 5, 8, 0, 4, 8, 5, 4, 6, 1, 0, 4, 0, 2, 1, 3, 7, 3, 4, 8, 9, 4, 1, 8, 0, 5, 4, 6, 6, 6, 6, 6, 6, 1, 2, 9, 8, 0, 8, 6, 8, 0, 5, 3, 9, 2, 5, 3, 6, 6, 8, 4, 8, 5, 7, 6, 2, 6, 1, 2, 8, 3, 5, 0, 3, 4, 3, 5, 5, 3, 0, 7, 2, 4, 8, 2, 2, 4, 4, 0, 3, 5, 1, 7, 6, 7, 7, 1
OFFSET
0,1
COMMENTS
Any two of the four constants {A337668, A337669, A371525, this} are algebraically independent over Q, while any three are not (Duverney et al., 2022).
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.
FORMULA
Equals Product_{k>=2} (1 - 1/A000032(k)).
Equals A371530 / (2*sqrt(5)).
Equals (phi^(1/4) / sqrt(5)) * 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).
EXAMPLE
0.33589766935710210314766572663122658048546104021373...
MATHEMATICA
With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[(Surd[GoldenRatio, 4] / Sqrt[5]) * eta[2*tau0]^2 * eta[6*tau0]/(eta[3*tau0] * eta[4*tau0]), 10, 120][[1]]]
PROG
(PARI) prodinf(k = 2, 1 - 1/(fibonacci(k-1) + fibonacci(k+1)))
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Mar 26 2024
STATUS
approved