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

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

Offset: 1

Author

Clark Kimberling, Mar 01 2018

Keywords

• nonn
• cons
• easy

Comments

The Lambert W function satisfies the functional equations

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)).

Guide to related constants:

--------------------------------------------

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

--------------------------------------------

1 1 A299613 A299614

1 2 A299615 A299616

1 e A030178 A299617

e e 2 exactly e^2 exactly

1 1/e A299618 A299619

1 3 A299620 A299621

1 1/2 A299622 A299623

1/2 1/2 A126583 A099954

2 2 A299624 A299625

3 3 A299626 A299627

1/3 1/3 A299628 A299629

3/2 3/2 A299630 A299631

e/2 e/2 A299632 A299633

Examples

			2*W(1) = 1.13428658081956774599993...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Lambert W-Function.
• Index entries for sequences related to LambertW function.
• Index entries for transcendental numbers.

Crossrefs

Cf. A299614-A299633.

Programs

Mathematica

w[x_] := ProductLog[x]; x = 1; y = 1; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299613 *)
RealDigits[2 ProductLog[1], 10, 111][[1]] (* Robert G. Wilson v, Mar 02 2018 *)

PARI

2*lambertw(1) \\ G. C. Greubel, Mar 07 2018

Formula

Equals 2*A030178.