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A202324 Decimal expansion of x < 0 satisfying x + 3 = exp(x).

%I A202324 #14 Nov 21 2024 15:37:52
%S A202324 2,9,4,7,5,3,0,9,0,2,5,4,2,2,8,5,1,2,7,5,9,0,1,2,6,3,8,8,7,1,3,9,8,1,
%T A202324 6,4,1,4,4,5,8,0,0,7,6,4,5,3,9,9,6,8,9,0,4,8,9,6,6,1,8,2,8,6,6,9,1,5,
%U A202324 6,3,9,3,7,8,3,2,2,1,1,0,0,2,3,9,5,4,7,7,7,6,5,5,4,3,8,9,1,5,3
%N A202324 Decimal expansion of x < 0 satisfying x + 3 = exp(x).
%C A202324 See A202320 for a guide to related sequences. The Mathematica program includes a graph.
%H A202324 G. C. Greubel, <a href="/A202324/b202324.txt">Table of n, a(n) for n = 1..10000</a>
%H A202324 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A202324 Equals -3 - LambertW(-exp(-3)). - _G. C. Greubel_, Nov 09 2017
%e A202324 x < 0: -2.9475309025422851275901263887139816414...
%e A202324 x > 0:  1.50524149579288336699862443213735394007...
%t A202324 u = 1; v = 3;
%t A202324 f[x_] := u*x + v; g[x_] := E^x
%t A202324 Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
%t A202324 r = x /. FindRoot[f[x] == g[x], {x, -2, -1}, WorkingPrecision -> 110]
%t A202324 RealDigits[r]  (* A202324 *)
%t A202324 r = x /. FindRoot[f[x] == g[x], {x, 1.5, 1.6}, WorkingPrecision -> 110]
%t A202324 RealDigits[r]  (* A202325 *)
%t A202324 RealDigits[-3 - LambertW[-Exp[-3]], 10, 100][[1]] (* _G. C. Greubel_, Nov 09 2017 *)
%o A202324 (PARI) solve(x=-3, 0, x+3-exp(x)) \\ _Michel Marcus_, Nov 09 2017
%Y A202324 Cf. A202320.
%K A202324 nonn,cons
%O A202324 1,1
%A A202324 _Clark Kimberling_, Dec 16 2011