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%I #9 Nov 24 2024 20:48:44
%S 1,3,7,0,1,5,3,8,8,4,3,0,9,1,8,7,8,9,2,0,5,6,4,9,8,9,6,1,0,7,5,2,6,0,
%T 3,7,6,8,2,8,1,1,1,4,3,1,3,6,1,6,4,1,0,6,7,0,8,1,9,6,0,3,0,9,9,7,5,0,
%U 0,7,7,5,7,0,2,2,3,7,6,2,9,5,6,2,3,9
%N Decimal expansion of 2*W(e/2), 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(e/2) = W(e^2/(2*W(e/2))) = 2 - log(4) - 2*log(W(e/2)). See A299613 for a guide to related sequences.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%e 2*W(e/2) = 1.3701538843091878920564989610752603...
%t w[x_] := ProductLog[x]; x = E/2; y = E/2; u = N[w[x] + w[y], 100]
%t RealDigits[u, 10][[1]] (* A299632 *)
%o (PARI) 2*lambertw(exp(1)/2) \\ _Altug Alkan_, Mar 13 2018
%Y Cf. A299613, A299632.
%K nonn,cons,easy
%O 0,2
%A _Clark Kimberling_, Mar 13 2018