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A199429
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Decimal expansion of x>0 satisfying x^2+x*sin(x)=cos(x).
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57
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6, 4, 3, 4, 3, 6, 3, 6, 4, 1, 3, 8, 0, 2, 6, 1, 5, 8, 6, 4, 2, 0, 9, 8, 9, 1, 4, 3, 0, 4, 0, 1, 3, 1, 8, 2, 6, 8, 7, 4, 4, 6, 7, 2, 4, 1, 9, 4, 5, 7, 8, 5, 1, 6, 3, 2, 3, 8, 7, 4, 9, 1, 9, 8, 5, 8, 8, 7, 5, 2, 2, 9, 2, 2, 2, 7, 2, 5, 9, 4, 1, 7, 6, 4, 1, 7, 8, 8, 8, 7, 0, 7, 8, 5, 2, 7, 8, 5, 7
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OFFSET
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0,1
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COMMENTS
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For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*sin(x)=c*cos(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
1.... 1.... 1.... A199429
1.... 1.... 2.... A199430
1.... 1.... 3.... A199431
1.... 2.... 1.... A199432
1.... 2.... 2.... A199433
1.... 2.... 3.... A199434
1.... 3.... 1.... A199435
1.... 3.... 2.... A199436
1.... 3.... 3.... A199437
2.... 1.... 1.... A199438
2.... 1.... 2.... A199439
2.... 1.... 3.... A199440
2.... 2.... 1.... A199441
2.... 2.... 3.... A199442
2.... 3.... 1.... A199443
2.... 3.... 2.... A199444
2.... 3.... 3.... A199445
2.... 1.... 1.... A199446
3.... 1.... 2.... A199447
3.... 1.... 3.... A199448
3.... 2.... 1.... A199449
3.... 2.... 2.... A199450
3.... 2.... 3.... A199451
3.... 3.... 1.... A199452
3.... 3.... 2.... A199453
1... -1.... 1.... A199454
1... -1.... 2.... A199455
1... -1.... 3.... A199456
1... -2... -3.... A199457
1... -2... -2.... A199458
1... -2... -1.... A199459
1... -2... 0.... A199460
1... -2... 1.... A199461
1... -2... 2.... A199462
1... -2... 3.... A199463
1... -3... -3.... A199464
1... -3... -2.... A199465
1... -3... -1.... A199466
1... -3... 0.... A199467
1... -3... 1.... A199468
1... -3... 2.... A199469
1... -3... 3.... A199470
2... -1... 1.... A199471
2... -1... 2.... A199472
2... -1... 3.... A199473
2... -2... 1.... A199503
2... -2... 3.... A199504
3... -1... 1.... A199505
2... -1... 2.... A199506
2... -1... 3.... A199507
2... -2... 1.... A199508
2... -2... 2.... A199509
2... -2... 3.... A199510
3... -3... 1.... A199511
3... -3... 2.... A199513
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 A199429, take f(x,u,v)=x^2+u*x*sin(x)-v*cos(x) 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.
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LINKS
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Table of n, a(n) for n=0..98.
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EXAMPLE
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x=0.6434363641380261586420989143040131826874...
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MATHEMATICA
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(* Program 1: A199429 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Sin[x]; g[x_] := c*Cos[x]
Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .64, .65}, WorkingPrecision -> 110]
RealDigits[r] (* A199429 *)
(* Program 2: implicit surface: x^2+u*x*sin(x)=v*cos(x) *)
f[{x_, u_, v_}] := x^2 + u*x*Sin[x] - v*Cos[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 10}, {v, u, 100}];
ListPlot3D[Flatten[t, 1]] (* for A199429 *)
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CROSSREFS
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Cf. A199370, A199170, A198866, A198755, A198414, A197737.
Sequence in context: A245634 A182618 A118227 * A235509 A224927 A200104
Adjacent sequences: A199426 A199427 A199428 * A199430 A199431 A199432
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KEYWORD
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nonn,cons
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AUTHOR
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Clark Kimberling, Nov 06 2011
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STATUS
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approved
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