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Decimal expansion of 2*W(1), where W is the Lambert W function (or PowerLog); see Comments.
21

%I #39 Jul 28 2020 21:58:34

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

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

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

%N Decimal expansion of 2*W(1), where W is the Lambert W function (or PowerLog); see Comments.

%C The Lambert W function satisfies the functional equations

%C W(x) + W(y) = W(x*y(1/W(x) + 1/W(y)) = log(x*y)/(W(x)*W(y)) for x and y greater than -1/e, so that 2*W(1) = W(2/W(1)) = -2*log(W(1)).

%C Guide to related constants:

%C --------------------------------------------

%C x y W(x) + W(y) e^(W(x) + W(y))

%C --------------------------------------------

%C 1 1 A299613 A299614

%C 1 2 A299615 A299616

%C 1 e A030178 A299617

%C e e 2 exactly e^2 exactly

%C 1 1/e A299618 A299619

%C 1 3 A299620 A299621

%C 1 1/2 A299622 A299623

%C 1/2 1/2 A126583 A099954

%C 2 2 A299624 A299625

%C 3 3 A299626 A299627

%C 1/3 1/3 A299628 A299629

%C 3/2 3/2 A299630 A299631

%C e/2 e/2 A299632 A299633

%H Clark Kimberling, <a href="/A299613/b299613.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>

%H <a href="/index/La#LambertW">Index entries for sequences related to LambertW function</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals 2*A030178.

%e 2*W(1) = 1.13428658081956774599993...

%t w[x_] := ProductLog[x]; x = 1; y = 1; u = N[w[x] + w[y], 100]

%t RealDigits[u, 10][[1]] (* A299613 *)

%t RealDigits[2 ProductLog[1], 10, 111][[1]] (* _Robert G. Wilson v_, Mar 02 2018 *)

%o (PARI) 2*lambertw(1) \\ _G. C. Greubel_, Mar 07 2018

%Y Cf. A299614-A299633.

%K nonn,cons,easy

%O 1,3

%A _Clark Kimberling_, Mar 01 2018