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A198755 Decimal expansion of x>0 satisfying x^2+cos(x)=2. 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.

Sequence in context: A035546 A182714 A262395 * A134237 A227192 A099889

Adjacent sequences:  A198752 A198753 A198754 * A198756 A198757 A198758

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 30 2011

STATUS

approved

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Last modified May 28 01:34 EDT 2020. Contains 334671 sequences. (Running on oeis4.)