A198361 -id:A198361 - 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

A196361 Decimal expansion of the absolute minimum of cos(t) + cos(2t) + cos(3t).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 28 2011

The function f(x) = cos(x) + cos(2x) + ... + cos(nx), where n >= 2, attains an absolute minimum at some c between 0 and Pi. Related sequences (with graphs in Mathematica programs):
  n      x     min(f(x))
  =   =======  =========
  2   A140244    -9/8
  3   A198670   A198361
  4   A198672   A198671
  5   A198674   A198673
  6   A198676   A198675

			x = 1.2929430585054266652256311954691354...
min(f(x)) = -1.3155651547204494123522707...

Idris Mercer, On a function related to Chowla's cosine problem, arXiv:1206.5012v1 [math.CA], June 21 2012.
Index entries for algebraic numbers, degree 2.

Cf. A198670.

n = 3; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = 2 Pi - t /. Part[x, 2]
RealDigits[u]   (* A196361 *)
RealDigits[v]   (* A198670 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
-(17 + 7*Sqrt[7])/27 // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Feb 19 2013 *)

(17+7*sqrt(7))/27 \\ Charles R Greathouse IV, Feb 07 2025

Equals (17+7*sqrt(7))/27. [Jonathan Vos Post, Jun 21 2012]

A198362 Decimal expansion of greatest x having 4*x^2+3x=cos(x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 24 2011

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

			least x: -0.91615106109683577000135072803946391...
greatest x: 0.244045322629135591466858282939448079493...

Cf. A197737.

a = 4; b = 3; 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, -1, -.9}, WorkingPrecision -> 110]
RealDigits[r1] (* A198361 *)
r2 = x /. FindRoot[f[x] == g[x], {x, .24, .25}, WorkingPrecision -> 110]
RealDigits[r2] (* A198362 *)

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

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A196361 Decimal expansion of the absolute minimum of cos(t) + cos(2t) + cos(3t).

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A198362 Decimal expansion of greatest x having 4*x^2+3x=cos(x).

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Mathematica