A202538 -id:A202538 - cp's OEIS frontend

A202537 Decimal expansion of x satisfying e^x-e^(-2x)=1.

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

Offset: 0

Clark Kimberling, Dec 21 2011

If u>0 and v>0, there is a unique number x satisfying e^(ux)-e^(-vx)=1.  Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 1.... A002390
1.... 2.... A202537
1.... 3.... A202538
2.... 1.... A202539
3.... 1.... A202540
2.... 2.... A202541
3.... 3.... A202542
1/2..1/2... A202543
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.  For an example related to A202537, take f(x,u,v)=e^(ux)-e^(-vx)-1 and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			0.382245085840035641329358499184857393759416422...

Index entries for transcendental numbers.

Cf. A002390.

(* Program 1:  A202537 *)
u = 1; v = 2;
f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .3, .4}, WorkingPrecision -> 110]
RealDigits[r]  (* A202537 *)
(* Program 2: implicit surface for e^(ux)-e(-vx)=1 *)
f[{x_, u_, v_}] := E^(u*x) - E^(-v*x) - 1;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, 1, 4}, {u, 2, 20}];
ListPlot3D[Flatten[t, 1]] (* for A202537 *)
First[ RealDigits[ Log[ Root[#^3 - #^2 - 1 & , 1]], 10, 99]] (* Jean-François Alcover, Feb 26 2013 *)

solve(x=0,1,exp(x)-exp(-2*x)-1) \\ Charles R Greathouse IV, Feb 26 2013

log(polrootsreal(x^3-x^2-1)[1]) \\ Charles R Greathouse IV, Feb 07 2025

Digits from a(90) on corrected by Jean-François Alcover, Feb 26 2013

A356032 Decimal expansion of the positive real root of x^4 + x - 1.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Aug 27 2022

The other real (negative) root is -A060007.
One of the pair of complex conjugate roots is obtained by negating sqrt(2*u) and sqrt(u) in the formula for r below, giving 0.248126062... - 1.033982060...*i.
Also, the absolute value of the negative real root of x^4 - x - 1, cf. A060007. - M. F. Hasler, Jul 12 2025

			r = 0.724491959000515611588372282187036565786494481350011017270...

Cf. A060007 (positive root of x^4 - x - 1), A072223, A086106, A202538, A376658.

First[RealDigits[x/.N[{x->Root[-1+#1+#1^4 &,2,0]},75]]] (* Stefano Spezia, Aug 27 2022 *)

solve(x=0, 1, x^4 + x - 1) \\ Michel Marcus, Aug 28 2022

polrootsreal(x^4 + x - 1)[2] \\ M. F. Hasler, Jul 12 2025

r = (-sqrt(2)*u + sqrt(sqrt(2*u) - 2*u^2))/(2*sqrt(u)), with u = (Ap^(1/3) + ep*Am^(1/3))/3, where Ap = (3/16)*(9 + sqrt(3*283)), Am = (3/16)*(9 - sqrt(3*283)) and ep = (-1 + sqrt(3)*i)/2, with i = sqrt(-1). For the trigonometric version set u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))).

Equals sqrt(A072223) = 1/A086106 = 1/exp(A202538). - Hugo Pfoertner, Jul 13 2025

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A202537 Decimal expansion of x satisfying e^x-e^(-2x)=1.

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A356032 Decimal expansion of the positive real root of x^4 + x - 1.

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