A199949 Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).

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

Offset

0,1

Comments

For many choices of a,b,c, there are exactly two numbers x>0 satisfying a*x^2+b*cos(x)=c*sin(x).

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

a.... b.... c.... least x, greatest x

1.... 1.... 2.... A199949, A199950

1.... 1.... 3.... A199951, A199952

1.... 1.... 4.... A199953, A199954

1.... 2.... 3.... A199955, A199956

1.... 2.... 4.... A199957, A199958

1.... 3.... 3.... A199959, A199960

1.... 3.... 4.... A199961, A199962

1.... 4.... 3.... A199963, A199964

1.... 4.... 4.... A199965, A199966

2.... 1.... 3.... A199967, A200003

2.... 1.... 4.... A200004, A200005

3.... 1.... 4.... A200006, A200007

4.... 1.... 4.... A200008, A200009

1... -1.... 1.... A200010, A200011

1... -1.... 2.... A200012, A200013

1... -1.... 3.... A200014, A200015

1... -1.... 4.... A200016, A200017

1... -2.... 1.... A200018, A200019

1... -2.... 2.... A200020, A200021

1... -2.... 3.... A200022, A200023

1... -2.... 4.... A200024, A200025

1... -3.... 1.... A200026, A200027

1... -3.... 2.... A200093, A200094

1... -3.... 3.... A200095, A200096

1... -3.... 4.... A200097, A200098

1... -4.... 1.... A200099, A200100

1... -4.... 2.... A200101, A200102

1... -4.... 3.... A200103, A200104

1... -4.... 4.... A200105, A200106

2... -1.... 1.... A200107, A200108

2... -1.... 2.... A200109, A200110

2... -1.... 3.... A200111, A200112

2... -1.... 4.... A200114, A200115

2... -2.... 1.... A200116, A200117

2... -2.... 3.... A200118, A200119

2... -3.... 1.... A200120, A200121

2... -3.... 2.... A200122, A200123

2... -3.... 3.... A200124, A200125

2... -3.... 4.... A200126, A200127

2... -4.... 1.... A200128, A200129

2... -4.... 3.... A200130, A200131

3... -1.... 1.... A200132, A200133

3... -1.... 2.... A200223, A200224

3... -1.... 3.... A200225, A200226

3... -1.... 4.... A200227, A200228

3... -2.... 1.... A200229, A200230

3... -2.... 2.... A200231, A200232

3... -2.... 3.... A200233, A200234

3... -2.... 4.... A200235, A200236

3... -3.... 1.... A200237, A200238

3... -3.... 2.... A200239, A200240

3... -3.... 4.... A200241, A200242

3... -4.... 1.... A200277, A200278

3... -4.... 2.... A200279, A200280

3... -4.... 3.... A200281, A200282

3... -4.... 4.... A200283, A200284

4... -1.... 1.... A200285, A200286

4... -1.... 2.... A200287, A200288

4... -1.... 3.... A200289, A200290

4... -1.... 4.... A200291, A200292

4... -2.... 1.... A200293, A200294

4... -2.... 3.... A200295, A200296

4... -3.... 1.... A200299, A200300

4... -3.... 2.... A200297, A200298

4... -3.... 3.... A200301, A200302

4... -3.... 4.... A200303, A200304

4... -4.... 1.... A200305, A200306

4... -4.... 3.... A200307, A200308

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 A199949, take f(x,u,v)=x^2+u*cos(x)-v*sin(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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Example

least x:  0.659266045766946074537348579563067611...
greatest x: 1.2710268008159460640047188480978502...

Mathematica

(* Program 1:  A199949 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .65, .66}, WorkingPrecision -> 110]
RealDigits[r]   (* A199949 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.27, 1.28}, WorkingPrecision -> 110]
RealDigits[r]   (* A199950 *)
(* Program 2: implicit surface of x^2+u*cos(x)=v*sin(x) *)
f[{x_, u_, v_}] := x^2 + u*Cos[x] - v*Sin[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, -5, 0}, {v, 0, 1}];
ListPlot3D[Flatten[t, 1]]  (* for A199949 *)

Prog

(PARI) a=1; b=1; c=2; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ G. C. Greubel, Jul 05 2018

Crossrefs

Cf. A199950.

Sequence in context: A275110 A011284 A196760 * A165227 A242761 A200477

Adjacent sequences: A199946 A199947 A199948 * A199950 A199951 A199952

Keyword

nonn,cons

Author

Clark Kimberling, Nov 12 2011

Extensions

A-number corrected by Jaroslav Krizek, Nov 27 2011

Status

approved