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Decimal expansion of x<0 having x^2+x=cos(x).
144

%I #21 Aug 05 2021 16:26:37

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

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

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

%N Decimal expansion of x<0 having x^2+x=cos(x).

%C For many choices of a,b,c, there are exactly two numbers x having a*x^2+b*x=cos(x).

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

%C a.... b.... c.... x

%C 1.... 0.... 1.... A125578

%C 1.... 0.... 2.... A197806

%C 1.... 0.... 3.... A197807

%C 1.... 0.... 4.... A197808

%C 1.... 1.... 1.... A197737, A197738

%C 1.... 1.... 2.... A197809, A197810

%C 1.... 1.... 3.... A197811, A197812

%C 1.... 1.... 4.... A197813, A197814

%C 1... -2... -1.... A197815, A197820

%C 1... -3... -1.... A197825, A197831

%C 1... -4... -1.... A197839, A197840

%C 1.... 2.... 1.... A197841, A197842

%C 1.... 2.... 2.... A197843, A197844

%C 1.... 2.... 3.... A197845, A197846

%C 1.... 2.... 4.... A197847, A197848

%C 1... -2... -2.... A197849, A197850

%C 1... -3... -2.... A198098, A198099

%C 1... -4... -2.... A198100, A198101

%C 1.... 3.... 1.... A198102, A198103

%C 1.... 3.... 2.... A198104, A198105

%C 1.... 3.... 3.... A198106, A198107

%C 1.... 3.... 4.... A198108, A198109

%C 1... -2... -3.... A198140, A198141

%C 1... -3... -3.... A198142, A198143

%C 1... -4... -3.... A198144, A198145

%C 2.... 0.... 1.... A198110

%C 2.... 0.... 3.... A198111

%C 2.... 1.... 1.... A198112, A198113

%C 2.... 1.... 2.... A198114, A198115

%C 2.... 1.... 3.... A198116, A198117

%C 2.... 1.... 4.... A198118, A198119

%C 2.... 1... -1.... A198120, A198121

%C 2... -4... -1.... A198122, A198123

%C 2.... 2.... 1.... A198124, A198125

%C 2.... 2.... 3.... A198126, A198127

%C 2.... 3.... 1.... A198128, A198129

%C 2.... 3.... 2.... A198130, A198131

%C 2.... 3.... 3.... A198132, A198133

%C 2.... 3.... 4.... A198134, A198135

%C 2... -4... -3.... A198136, A198137

%C 3.... 0.... 1.... A198211

%C 3.... 0.... 2.... A198212

%C 3.... 0.... 4.... A198213

%C 3.... 1.... 1.... A198214, A198215

%C 3.... 1.... 2.... A198216, A198217

%C 3.... 1.... 3.... A198218, A198219

%C 3.... 1.... 4.... A198220, A198221

%C 3.... 2.... 1.... A198222, A198223

%C 3.... 2.... 2.... A198224, A198225

%C 3.... 2.... 3.... A198226, A198227

%C 3.... 2.... 4.... A198228, A198229

%C 3.... 3.... 1.... A198230, A198231

%C 3.... 3.... 2.... A198232, A198233

%C 3.... 3.... 4.... A198234, A198235

%C 3.... 4.... 1.... A198236, A198237

%C 3.... 4.... 2.... A198238, A198239

%C 3.... 4.... 3.... A198240, A198241

%C 3.... 4.... 4.... A198138, A198139

%C 3... -4... -1.... A198345, A198346

%C 4.... 0.... 1.... A198347

%C 4.... 0.... 3.... A198348

%C 4.... 1.... 1.... A198349, A198350

%C 4.... 1.... 2.... A198351, A198352

%C 4.... 1.... 3.... A198353, A198354

%C 4.... 1.... 4.... A198355, A198356

%C 4.... 2.... 1.... A198357, A198358

%C 4.... 2.... 3.... A198359, A198360

%C 4.... 3.... 1.... A198361, A198362

%C 4.... 3.... 2.... A198363, A198364

%C 4.... 3.... 3.... A198365, A198366

%C 4.... 3.... 4.... A198367, A198368

%C 4.... 4.... 1.... A198369, A198370

%C 4.... 4.... 3.... A198371, A198372

%C 4... -4... -1.... A198373, A198374

%C 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.

%C For an example related to A197737, take f(x,u,v)=x^2+u*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.

%e negative: -1.25115183522076481159287006878816185994...

%e positive: 0.55000934992726156666495361947172926116...

%t (* Program 1: A197738 *)

%t a = 1; b = 1; c = 1;

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

%t Plot[{f[x], g[x]}, {x, -2, 1}]

%t r1 = x /. FindRoot[f[x] == g[x], {x, -1.26, -1.25}, WorkingPrecision -> 110]

%t RealDigits[r1] (* A197737 *)

%t r1 = x /. FindRoot[f[x] == g[x], {x, .55, .551}, WorkingPrecision -> 110]

%t RealDigits[r1] (* A197738 *)

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

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

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

%t ListPlot3D[Flatten[t, 1]] (* for A197737 *)

%o (PARI) A197737_vec(N=150)={localprec(N+10); digits(solve(x=-1.5,-1,x^2+x-cos(x))\.1^N)} \\ _M. F. Hasler_, Aug 05 2021

%Y Cf. A197738.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Oct 20 2011