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%I #24 Jul 06 2018 17:10:04
%S 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,
%T 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,
%U 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
%N Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).
%C For many choices of a,b,c, there are exactly two numbers x>0 satisfying a*x^2+b*cos(x)=c*sin(x).
%C Guide to related sequences, with graphs included in Mathematica programs:
%C a.... b.... c.... least x, greatest x
%C 1.... 1.... 2.... A199949, A199950
%C 1.... 1.... 3.... A199951, A199952
%C 1.... 1.... 4.... A199953, A199954
%C 1.... 2.... 3.... A199955, A199956
%C 1.... 2.... 4.... A199957, A199958
%C 1.... 3.... 3.... A199959, A199960
%C 1.... 3.... 4.... A199961, A199962
%C 1.... 4.... 3.... A199963, A199964
%C 1.... 4.... 4.... A199965, A199966
%C 2.... 1.... 3.... A199967, A200003
%C 2.... 1.... 4.... A200004, A200005
%C 3.... 1.... 4.... A200006, A200007
%C 4.... 1.... 4.... A200008, A200009
%C 1... -1.... 1.... A200010, A200011
%C 1... -1.... 2.... A200012, A200013
%C 1... -1.... 3.... A200014, A200015
%C 1... -1.... 4.... A200016, A200017
%C 1... -2.... 1.... A200018, A200019
%C 1... -2.... 2.... A200020, A200021
%C 1... -2.... 3.... A200022, A200023
%C 1... -2.... 4.... A200024, A200025
%C 1... -3.... 1.... A200026, A200027
%C 1... -3.... 2.... A200093, A200094
%C 1... -3.... 3.... A200095, A200096
%C 1... -3.... 4.... A200097, A200098
%C 1... -4.... 1.... A200099, A200100
%C 1... -4.... 2.... A200101, A200102
%C 1... -4.... 3.... A200103, A200104
%C 1... -4.... 4.... A200105, A200106
%C 2... -1.... 1.... A200107, A200108
%C 2... -1.... 2.... A200109, A200110
%C 2... -1.... 3.... A200111, A200112
%C 2... -1.... 4.... A200114, A200115
%C 2... -2.... 1.... A200116, A200117
%C 2... -2.... 3.... A200118, A200119
%C 2... -3.... 1.... A200120, A200121
%C 2... -3.... 2.... A200122, A200123
%C 2... -3.... 3.... A200124, A200125
%C 2... -3.... 4.... A200126, A200127
%C 2... -4.... 1.... A200128, A200129
%C 2... -4.... 3.... A200130, A200131
%C 3... -1.... 1.... A200132, A200133
%C 3... -1.... 2.... A200223, A200224
%C 3... -1.... 3.... A200225, A200226
%C 3... -1.... 4.... A200227, A200228
%C 3... -2.... 1.... A200229, A200230
%C 3... -2.... 2.... A200231, A200232
%C 3... -2.... 3.... A200233, A200234
%C 3... -2.... 4.... A200235, A200236
%C 3... -3.... 1.... A200237, A200238
%C 3... -3.... 2.... A200239, A200240
%C 3... -3.... 4.... A200241, A200242
%C 3... -4.... 1.... A200277, A200278
%C 3... -4.... 2.... A200279, A200280
%C 3... -4.... 3.... A200281, A200282
%C 3... -4.... 4.... A200283, A200284
%C 4... -1.... 1.... A200285, A200286
%C 4... -1.... 2.... A200287, A200288
%C 4... -1.... 3.... A200289, A200290
%C 4... -1.... 4.... A200291, A200292
%C 4... -2.... 1.... A200293, A200294
%C 4... -2.... 3.... A200295, A200296
%C 4... -3.... 1.... A200299, A200300
%C 4... -3.... 2.... A200297, A200298
%C 4... -3.... 3.... A200301, A200302
%C 4... -3.... 4.... A200303, A200304
%C 4... -4.... 1.... A200305, A200306
%C 4... -4.... 3.... A200307, A200308
%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 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.
%H G. C. Greubel, <a href="/A199949/b199949.txt">Table of n, a(n) for n = 0..10000</a>
%e least x: 0.659266045766946074537348579563067611...
%e greatest x: 1.2710268008159460640047188480978502...
%t (* Program 1: A199949 *)
%t a = 1; b = 1; c = 2;
%t f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, .65, .66}, WorkingPrecision -> 110]
%t RealDigits[r] (* A199949 *)
%t r = x /. FindRoot[f[x] == g[x], {x, 1.27, 1.28}, WorkingPrecision -> 110]
%t RealDigits[r] (* A199950 *)
%t (* Program 2: implicit surface of x^2+u*cos(x)=v*sin(x) *)
%t f[{x_, u_, v_}] := x^2 + u*Cos[x] - v*Sin[x];
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, -5, 0}, {v, 0, 1}];
%t ListPlot3D[Flatten[t, 1]] (* for A199949 *)
%o (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
%Y Cf. A199950.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Nov 12 2011
%E A-number corrected by _Jaroslav Krizek_, Nov 27 2011
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