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A201939 Decimal expansion of x>0 satisfying x*cosh(x)=2. 10
1, 1, 5, 0, 5, 8, 4, 9, 6, 7, 4, 1, 8, 6, 6, 3, 9, 4, 9, 5, 3, 4, 9, 3, 3, 7, 3, 3, 6, 1, 3, 7, 8, 8, 1, 9, 5, 7, 6, 6, 8, 3, 7, 4, 9, 4, 8, 4, 4, 2, 4, 2, 3, 4, 1, 1, 8, 3, 3, 9, 2, 5, 1, 8, 0, 8, 8, 3, 2, 2, 5, 4, 6, 1, 7, 6, 4, 1, 1, 7, 2, 8, 0, 1, 3, 6, 7, 5, 4, 4, 1, 4, 5, 2, 4, 6, 9, 9, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For many choices of u and v, there is exactly one x>0 satisfying x*cosh(u*x)=v.  Guide to related sequences, with graphs included in Mathematica programs:

u.... v.... x

1.... 1.... A069814

1.... 2.... A201939

1.... 3.... A201943

2.... 1.... A201944

3.... 1.... A201945

2.... 2.... A202283

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 A201939, take f(x,u,v)=x*cosh(u*x)-v 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.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=1.15058496741866394953493373361378819576...

MATHEMATICA

(* Program 1:  A201939 *)

u = 1; v = 2;

f[x_] := x*Cosh[u*x]; g[x_] := v

Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]

r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]

RealDigits[r]  (* A201939 *)

(* Program 2: implicit surface of u*cosh(x)=v *)

f[{x_, u_, v_}] := x*Cosh[u*x] - v;

t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 20}, {u, 1, 9}];

ListPlot3D[Flatten[t, 1]] (* for A201939 *)

CROSSREFS

Cf. A201946.

Sequence in context: A265302 A228764 A200631 * A256192 A154814 A341482

Adjacent sequences:  A201936 A201937 A201938 * A201940 A201941 A201942

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Dec 15 2011

STATUS

approved

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Last modified June 16 16:09 EDT 2021. Contains 345063 sequences. (Running on oeis4.)