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%I #18 Feb 20 2019 21:44:41
%S 1,4,0,9,6,2,4,0,0,4,0,0,2,5,9,6,2,4,9,2,3,5,5,9,3,9,7,0,5,8,9,4,9,3,
%T 5,4,7,1,2,3,5,4,8,3,5,1,0,7,8,9,0,1,5,1,5,1,0,1,6,6,8,3,0,0,9,9,1,8,
%U 3,6,0,1,6,7,3,1,8,1,4,5,2,5,1,6,8,7,4,8,9,2,1,4,3,2,5,9,0,7,9
%N Decimal expansion of x < 0 satisfying x^2 + sin(x) = 1.
%C For many choices of a,b,c, there are exactly two numbers x having a*x^2 + b*sin(x) = c.
%C Guide to related sequences, with graphs included in Mathematica programs:
%C a.... b.... c.... x
%C 1.... 1.... 1.... A124597
%C 1.... 1.... 1.... A198866, A198867
%C 1.... 1.... 2.... A199046, A199047
%C 1.... 1.... 3.... A199048, A199049
%C 1.... 2.... 0.... A198414
%C 1.... 2.... 1.... A199080, A199081
%C 1.... 2.... 2.... A199082, A199083
%C 1.... 2.... 3.... A199050, A199051
%C 1.... 3.... 0.... A198415
%C 1.... 3... -1.... A199052, A199053
%C 1.... 3.... 1.... A199054, A199055
%C 1.... 3.... 2.... A199056, A199057
%C 1.... 3.... 3.... A199058, A199059
%C 2.... 1.... 0.... A198583
%C 2.... 1.... 1.... A199061, A199062
%C 2.... 1.... 2.... A199063, A199064
%C 2.... 1.... 3.... A199065, A199066
%C 2.... 2.... 1.... A199067, A199068
%C 2.... 2.... 3.... A199069, A199070
%C 2.... 3.... 0.... A198605
%C 2.... 3.... 1.... A199071, A199072
%C 2.... 3.... 2.... A199073, A199074
%C 2.... 3.... 3.... A199075, A199076
%C 3.... 0.... 1.... A020760
%C 3.... 1.... 1.... A199060, A199077
%C 3.... 1.... 2.... A199078, A199079
%C 3.... 1.... 3.... A199150, A199151
%C 3.... 2.... 1.... A199152, A199153
%C 3.... 2.... 2.... A199154, A199155
%C 3.... 2.... 3.... A199156, A199157
%C 3.... 3.... 1.... A199158, A199159
%C 3.... 3.... 2.... A199160, A199161
%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 A198866, take f(x,u,v) = x^2 + u*sin(x) - v 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="/A198866/b198866.txt">Table of n, a(n) for n = 1..10000</a>
%e negative: -1.40962400400259624923559397058949354...
%e positive: 0.63673265080528201088799090383828005...
%t (* Program 1: this sequence and A198867 *)
%t a = 1; b = 1; c = 1;
%t f[x_] := a*x^2 + b*Sin[x]; g[x_] := c
%t Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, -1.41, -1.40}, WorkingPrecision -> 110]
%t RealDigits[r] (* this sequence *)
%t r = x /. FindRoot[f[x] == g[x], {x, .63, .64}, WorkingPrecision -> 110]
%t RealDigits[r] (* A198867 *)
%t (* Program 2: implicit surface of x^2+u*sin(x)=v *)
%t f[{x_, u_, v_}] := x^2 + u*Sin[x] - v;
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 6}, {v, u, 12}];
%t ListPlot3D[Flatten[t, 1]] (* for this sequence *)
%o (PARI) a=1; b=1; c=1; solve(x=-2, 0, a*x^2 + b*sin(x) - c) \\ _G. C. Greubel_, Feb 20 2019
%o (Sage) a=1; b=1; c=1; (a*x^2 + b*sin(x)==c).find_root(-2,0,x) # _G. C. Greubel_, Feb 20 2019
%Y Cf. A198867, A198755, A198414, A197737.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Nov 02 2011
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