A198374 -id:A198374 - cp's OEIS frontend

A197737 Decimal expansion of x<0 having x^2+x=cos(x).

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, 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, 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

Offset: 1

Clark Kimberling, Oct 20 2011

For many choices of a,b,c, there are exactly two numbers x having a*x^2+b*x=cos(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 0.... 1.... A125578
... 0.... 2.... A197806
... 0.... 3.... A197807
... 0.... 4.... A197808
... 1.... 1.... A197737, A197738
... 1.... 2.... A197809, A197810
... 1.... 3.... A197811, A197812
... 1.... 4.... A197813, A197814
.. -2... -1.... A197815, A197820
.. -3... -1.... A197825, A197831
.. -4... -1.... A197839, A197840
... 2.... 1.... A197841, A197842
... 2.... 2.... A197843, A197844
... 2.... 3.... A197845, A197846
... 2.... 4.... A197847, A197848
.. -2... -2.... A197849, A197850
.. -3... -2.... A198098, A198099
.. -4... -2.... A198100, A198101
... 3.... 1.... A198102, A198103
... 3.... 2.... A198104, A198105
... 3.... 3.... A198106, A198107
... 3.... 4.... A198108, A198109
.. -2... -3.... A198140, A198141
.. -3... -3.... A198142, A198143
.. -4... -3.... A198144, A198145
... 0.... 1.... A198110
... 0.... 3.... A198111
... 1.... 1.... A198112, A198113
... 1.... 2.... A198114, A198115
... 1.... 3.... A198116, A198117
... 1.... 4.... A198118, A198119
... 1... -1.... A198120, A198121
.. -4... -1.... A198122, A198123
... 2.... 1.... A198124, A198125
... 2.... 3.... A198126, A198127
... 3.... 1.... A198128, A198129
... 3.... 2.... A198130, A198131
... 3.... 3.... A198132, A198133
... 3.... 4.... A198134, A198135
.. -4... -3.... A198136, A198137
... 0.... 1.... A198211
... 0.... 2.... A198212
... 0.... 4.... A198213
... 1.... 1.... A198214, A198215
... 1.... 2.... A198216, A198217
... 1.... 3.... A198218, A198219
... 1.... 4.... A198220, A198221
... 2.... 1.... A198222, A198223
... 2.... 2.... A198224, A198225
... 2.... 3.... A198226, A198227
... 2.... 4.... A198228, A198229
... 3.... 1.... A198230, A198231
... 3.... 2.... A198232, A198233
... 3.... 4.... A198234, A198235
... 4.... 1.... A198236, A198237
... 4.... 2.... A198238, A198239
... 4.... 3.... A198240, A198241
... 4.... 4.... A198138, A198139
.. -4... -1.... A198345, A198346
... 0.... 1.... A198347
... 0.... 3.... A198348
... 1.... 1.... A198349, A198350
... 1.... 2.... A198351, A198352
... 1.... 3.... A198353, A198354
... 1.... 4.... A198355, A198356
... 2.... 1.... A198357, A198358
... 2.... 3.... A198359, A198360
... 3.... 1.... A198361, A198362
... 3.... 2.... A198363, A198364
... 3.... 3.... A198365, A198366
... 3.... 4.... A198367, A198368
... 4.... 1.... A198369, A198370
... 4.... 3.... A198371, A198372
.. -4... -1.... A198373, A198374
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 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.

			negative: -1.25115183522076481159287006878816185994...
positive:  0.55000934992726156666495361947172926116...

Index entries for transcendental numbers.

Cf. A197738.

(* Program 1:  A197738 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x; g[x_] := c*Cos[x]
Plot[{f[x], g[x]}, {x, -2, 1}]
r1 = x /. FindRoot[f[x] == g[x], {x, -1.26, -1.25}, WorkingPrecision -> 110]
RealDigits[r1] (* A197737 *)
r1 = x /. FindRoot[f[x] == g[x], {x, .55, .551}, WorkingPrecision -> 110]
RealDigits[r1] (* A197738 *)
(* Program 2: implicit surface of x^2+u*x=v*cos(x) *)
f[{x_, u_, v_}] := x^2 + u*x - v*Cos[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 20}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]]  (* for A197737 *)

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

A198373 Decimal expansion of least x having 4*x^2-4x=cos(x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 24 2011

See A197737 for a guide to related sequences. The Mathematica program includes a graph.

			least x: 0.369948170589866528691602088620295223...
greatest x: 0.7633945382101923369926745363848376145...

Cf. A197737.

a = 4; b = -4; c = -1;
f[x_] := a*x^2 + b*x; g[x_] := c*Cos[x]
Plot[{f[x], g[x]}, {x, -1, 1}]
r1 = x /. FindRoot[f[x] == g[x], {x, .36, .37}, WorkingPrecision -> 110]
RealDigits[r1] (* A198373 *)
r2 = x /. FindRoot[f[x] == g[x], {x, .76, .77}, WorkingPrecision -> 110]
RealDigits[r2] (* A198374 *)

cp's OEIS Frontend

A197737 Decimal expansion of x<0 having x^2+x=cos(x).

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