%I #22 Jan 05 2025 19:51:37

%S 5,9,5,2,4,1,4,3,9,5,7,7,7,1,1,1,0,9,0,1,8,0,3,0,8,2,0,7,7,4,2,5,1,7,

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

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

%N Decimal expansion of phi/e, where phi = (1+sqrt(5))/2.

%H Ivan Panchenko, <a href="/A094880/b094880.txt">Table of n, a(n) for n = 0..1000</a>

%H Corey Martinsen and Pantelimon Stănică, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/53-3/MartinsenStanica1182015.pdf">Asymptotic Behavior of Gaps Between Roots of Weighted Factorials</a>, Fibonacci Quart. 53 (2015), no. 3, 213-218. See Corollary 2.4 p. 216.

%F From _Michel Marcus_, Jan 11 2022: (Start)

%F Equals A001622/A001113.

%F Equals lim_{n->oo} ((n+1)!*F(n+1))^(1/(n+1)) - (n!*F(n))^(1/n).

%F Equals lim_{n->oo} ((n+1)!*L(n+1))^(1/(n+1)) - (n!*L(n))^(1/n). (End)

%e 0.59524143957771110901803082077425172857166421077832...

%t RealDigits[GoldenRatio/E, 10, 105][[1]] (* _Amiram Eldar_, May 02 2023 *)

%o (PARI) ((1+sqrt(5))/2)/exp(1) \\ _Michel Marcus_, Jan 11 2022

%Y Cf. A001113 (e), A001622 (phi), A094868 (reciprocal).

%Y Cf. A000032 (Lucas), A000045 (Fibonacci).

%K cons,nonn

%O 0,1

%A _N. J. A. Sloane_, Jun 15 2004