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Decimal expansion of W(1) + W(1/2), where W is the Lambert W function (or PowerLog); see Comments.
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%I #18 Nov 24 2024 08:24:53

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

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

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

%N Decimal expansion of W(1) + W(1/2), 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 W(1) + W(1/2) = W((1/2)*(1/W(1) + 1/W(1/2))) = -log(2) - log(W(1)) - log(W(1/2)). See A299613 for a guide to related sequences.

%H G. C. Greubel, <a href="/A299622/b299622.txt">Table of n, a(n) for n = 0..10000</a>

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

%e W(1) + W(1/2) = 0.918877001658979699024877963140306614925280002...

%t w[x_] := ProductLog[x]; x = 1; y = 1/2; u = N[w[x] + w[y], 100]

%t RealDigits[u, 10][[1]] (* A299622 *)

%t RealDigits[LambertW[1] + LambertW[1/2], 10, 100][[1]] (* _G. C. Greubel_, Mar 03 2018 *)

%o (PARI) lambertw(1) + lambertw(1/2) \\ _G. C. Greubel_, Mar 03 2018

%Y Cf. A299613, A299623.

%K nonn,cons,easy

%O 0,1

%A _Clark Kimberling_, Mar 03 2018