A198822 -id:A198822

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A198755

Decimal expansion of x>0 satisfying x^2+cos(x)=2.

+10
106

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For many choices of a,b,c, there is a unique x>0 satisfying a*x^2+b*cos(x)=c.

Guide to related sequences, with graphs included in Mathematica programs:

a.... b.... c..... x

1.... 1.... 2..... A198755

1.... 1.... 3..... A198756

1.... 1.... 4..... A198757

1.... 2.... 3..... A198758

1.... 2.... 4..... A198811

1.... 3.... 3..... A198812

1.... 3.... 4..... A198813

1.... 4.... 3..... A198814

1.... 4.... 4..... A198815

1.... 1.... 0..... A125578

1... -1.... 1..... A198816

1... -1.... 2..... A198817

1... -1.... 3..... A198818

1... -1.... 4..... A198819

1... -2.... 1..... A198821

1... -2.... 2..... A198822

1... -2.... 3..... A198823

1... -2.... 4..... A198824

1... -2... -1..... A198825

1... -3.... 0..... A197807

1... -3.... 1..... A198826

1... -3.... 2..... A198828

1... -3.... 3..... A198829

1... -3.... 4..... A198830

1... -3... -1..... A198835

1... -3... -2..... A198836

1... -4.... 0..... A197808

1... -4.... 1..... A198838

1... -4.... 2..... A198839

1... -4.... 3..... A198840

1... -4.... 4..... A198841

1... -4... -1..... A198842

1... -4... -2..... A198843

1... -4... -3..... A198844

2.... 0.... 1..... A010503

2.... 0.... 3..... A115754

2.... 1.... 2..... A198820

2.... 1.... 3..... A198827

2.... 1.... 4..... A198837

2.... 2.... 3..... A198869

2.... 3.... 4..... A198870

2... -1.... 1..... A198871

2... -1.... 2..... A198872

2... -1.... 3..... A198873

2... -1.... 4..... A198874

2... -2... -1..... A198875

2... -2.... 3..... A198876

2... -3... -2..... A198877

2... -3... -1..... A198878

2... -3.... 1..... A198879

2... -3.... 2..... A198880

2... -3.... 3..... A198881

2... -3.... 4..... A198882

2... -4... -3..... A198883

2... -4... -1..... A198884

2... -4.... 1..... A198885

2... -4.... 3..... A198886

3.... 0.... 1..... A020760

3.... 1.... 2..... A198868

3.... 1.... 3..... A198917

3.... 1.... 4..... A198918

3.... 2.... 3..... A198919

3.... 2.... 4..... A198920

3.... 3.... 4..... A198921

3... -1.... 1..... A198922

3... -1.... 2..... A198924

3... -1.... 3..... A198925

3... -1.... 4..... A198926

3... -2... -1..... A198927

3... -2.... 1..... A198928

3... -2.... 2..... A198929

3... -2.... 3..... A198930

3... -2.... 4..... A198931

3... -3... -1..... A198932

3... -3.... 1..... A198933

3... -3.... 2..... A198934

3... -3.... 4..... A198935

3... -4... -3..... A198936

3... -4... -2..... A198937

3... -4... -1..... A198938

3... -4.... 1..... A198939

3... -4.... 2..... A198940

3... -4.... 3..... A198941

3... -4.... 4..... A198942

4.... 1.... 2..... A198923

4.... 1.... 3..... A198983

4.... 1.... 4..... A198984

4.... 2.... 3..... A198985

4.... 3.... 4..... A198986

4... -1.... 1..... A198987

4... -1.... 2..... A198988

4... -1.... 3..... A198989

4... -1.... 4..... A198990

4... -2... -1..... A198991

4... -2.... 1..... A198992

4... -2... -3..... A198993

4... -3... -2..... A198994

4... -3... -1..... A198995

4... -2.... 1..... A198996

4... -3.... 2..... A198997

4... -3.... 3..... A198998

4... -3.... 4..... A198999

4... -4... -3..... A199000

4... -4... -1..... A199001

4... -4.... 1..... A199002

4... -4.... 3..... A199003

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 A198755, take f(x,u,v)=x^2+u*cos(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=1.32562251814753662348322902938798744330...

MATHEMATICA

(* Program 1: A198655 *)

a = 1; b = 1; c = 2;

f[x_] := a*x^2 + b*Cos[x]; g[x_] := c

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

r = x /. FindRoot[f[x] == g[x], {x, 1.32, 1.33}, WorkingPrecision -> 110]

RealDigits[r] (* A198755 *)

(* Program 2: implicit surface of x^2+u*cos(x)=v *)

f[{x_, u_, v_}] := x^2 + u*Cos[x] - v;

t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 3}]}, {u, -5, 4}, {v, u, 20}];

ListPlot3D[Flatten[t, 1]] (* for A198755 *)

CROSSREFS

Cf. A197737, A198414.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 30 2011

STATUS

approved

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