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A199949
Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).
137
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
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
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 12 2011
EXTENSIONS
A-number corrected by Jaroslav Krizek, Nov 27 2011
STATUS
approved