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A299613 Decimal expansion of 2*W(1), where W is the Lambert W function (or PowerLog); see Comments.

%I A299613 #45 Nov 24 2024 08:25:02
%S A299613 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 A299613 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 A299613 9,1,5,8,6,1,7,3,3,6,9,1,3,3,3,8,6,4
%N A299613 Decimal expansion of 2*W(1), where W is the Lambert W function (or PowerLog); see Comments.
%C A299613 The Lambert W function satisfies the functional equations
%C A299613 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 A299613 Guide to related constants:
%C A299613 --------------------------------------------
%C A299613 x        y      W(x) + W(y)  e^(W(x) + W(y))
%C A299613 --------------------------------------------
%C A299613 1        1        A299613        A299614
%C A299613 1        2        A299615        A299616
%C A299613 1        e        A030178        A299617
%C A299613 e        e        2 exactly      e^2 exactly
%C A299613 1        1/e      A299618        A299619
%C A299613 1        3        A299620        A299621
%C A299613 1        1/2      A299622        A299623
%C A299613 1/2      1/2      A126583        A099954
%C A299613 2        2        A299624        A299625
%C A299613 3        3        A299626        A299627
%C A299613 1/3      1/3      A299628        A299629
%C A299613 3/2      3/2      A299630        A299631
%C A299613 e/2      e/2      A299632        A299633
%H A299613 Clark Kimberling, <a href="/A299613/b299613.txt">Table of n, a(n) for n = 1..10000</a>
%H A299613 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H A299613 <a href="/index/La#LambertW">Index entries for sequences related to LambertW function</a>.
%H A299613 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A299613 Equals 2*A030178.
%e A299613 2*W(1) = 1.13428658081956774599993...
%t A299613 w[x_] := ProductLog[x]; x = 1; y = 1; u = N[w[x] + w[y], 100]
%t A299613 RealDigits[u, 10][[1]]  (* A299613 *)
%t A299613 RealDigits[2 ProductLog[1], 10, 111][[1]] (* _Robert G. Wilson v_, Mar 02 2018 *)
%o A299613 (PARI) 2*lambertw(1) \\ _G. C. Greubel_, Mar 07 2018
%Y A299613 Cf. A299614-A299633.
%K A299613 nonn,cons,easy
%O A299613 1,3
%A A299613 _Clark Kimberling_, Mar 01 2018