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A199370 Decimal expansion of x>0 satisfying x^2+x*sin(x)=1.

%I A199370 #19 Feb 03 2025 21:29:40
%S A199370 7,2,2,5,8,7,5,4,9,9,2,2,5,2,4,7,6,8,3,5,5,9,3,2,8,7,2,8,7,7,1,9,6,7,
%T A199370 5,5,1,5,9,6,4,5,9,2,1,1,4,3,9,4,2,6,9,8,0,7,7,6,5,1,4,7,6,0,2,5,9,0,
%U A199370 9,4,2,5,0,7,3,1,6,0,1,8,3,0,3,4,3,5,6,2,9,4,1,8,7,2,7,9,8,3,3
%N A199370 Decimal expansion of x>0 satisfying x^2+x*sin(x)=1.
%C A199370 For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*sin(x)=c.
%C A199370 Guide to related sequences, with graphs included in Mathematica programs:
%C A199370 a.... b.... c.... x
%C A199370 1.... 1.... 1.... A199370
%C A199370 1.... 1.... 2.... A199371
%C A199370 1.... 1.... 3.... A199372
%C A199370 1.... 2.... 1.... A199373
%C A199370 1.... 2.... 2.... A199374
%C A199370 1.... 2.... 3.... A199375
%C A199370 1.... 3.... 1.... A199376
%C A199370 1.... 3.... 2.... A199377
%C A199370 1.... 3.... 3.... A199378
%C A199370 2.... 1.... 1.... A199379
%C A199370 2.... 1.... 2.... A199180
%C A199370 2.... 1.... 3.... A199181
%C A199370 2.... 2.... 1.... A199182
%C A199370 2.... 2.... 3.... A199183
%C A199370 2.... 3.... 1.... A199184
%C A199370 2.... 3.... 2.... A199185
%C A199370 2.... 3.... 3.... A199186
%C A199370 2.... 1.... 1.... A199187
%C A199370 3.... 1.... 2.... A199188
%C A199370 3.... 1.... 3.... A199189
%C A199370 3.... 2.... 1....
%C A199370 3.... 2.... 2....
%C A199370 3.... 2.... 3....
%C A199370 3.... 3.... 1....
%C A199370 3.... 3.... 2....
%C A199370 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 A199370 For an example related to A199370, take f(x,u,v)=x^2+u*x*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 A199370 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A199370 0.722587549922524768355932872877196755159...
%t A199370 (* Program 1: A199370 *)
%t A199370 a = 1; b = 1; c = 1;
%t A199370 f[x_] := a*x^2 + b*x*Sin[x]; g[x_] := c
%t A199370 Plot[{f[x], g[x]}, {x, -1, Pi}, {AxesOrigin -> {0, 0}}]
%t A199370 r = x /. FindRoot[f[x] == g[x], {x, .72, .73}, WorkingPrecision -> 110]
%t A199370 RealDigits[r]  (* A199370 *)
%t A199370 (* Program 2: implicit surface of x^2+u*x*sin(x)=v *)
%t A199370 f[{x_, u_, v_}] := x^2 + u*x*Sin[x] - v;
%t A199370 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 2.9}, {v, u, 600}];
%t A199370 ListPlot3D[Flatten[t, 1]]  (* for A199370 *)
%Y A199370 Cf. A199371, A199170, A198866, A198755, A198414, A197737.
%K A199370 nonn,cons
%O A199370 0,1
%A A199370 _Clark Kimberling_, Nov 05 2011