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%I #14 Mar 14 2018 06:38:33
%S 5,1,5,2,5,5,3,0,6,0,9,9,4,7,3,4,0,8,5,6,5,8,3,2,4,0,3,2,5,2,1,9,5,5,
%T 8,1,8,1,9,3,8,5,2,9,5,0,0,6,4,0,8,9,8,3,0,6,7,9,0,2,2,8,8,1,3,2,6,3,
%U 8,2,5,8,5,5,0,4,0,8,7,4,4,9,1,9,2,7
%N Decimal expansion of 2*W(1/3), 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(1/3) = W(2)/(9*W(1/3)) = -2*log(3) - 2*log(W(1/3)). See A299613 for a guide to related sequences.
%H G. C. Greubel, <a href="/A299628/b299628.txt">Table of n, a(n) for n = 0..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>
%e 2*W(1/3) = 0.5152553060994734085658324032521955...
%t w[x_] := ProductLog[x]; x = 1/3; y = 1/3; u = N[w[x] + w[y], 100]
%t RealDigits[u, 10][[1]] (* A299628 *)
%t RealDigits[2*LambertW[1/3], 10, 100][[1]] (* _G. C. Greubel_, Mar 06 2018 *)
%o (PARI) 2*lambertw(1/3) \\ _G. C. Greubel_, Mar 06 2018
%Y Cf. A299613, A299629.
%K nonn,cons,easy
%O 0,1
%A _Clark Kimberling_, Mar 03 2018
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