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%I #19 Sep 12 2021 12:53:27
%S 1,1,8,8,1,8,5,1,3,4,4,5,1,4,3,8,8,0,3,2,1,7,8,1,0,9,7,2,9,0,7,6,5,2,
%T 5,9,7,3,8,3,2,4,2,5,6,1,2,8,4,1,4,7,1,9,4,1,8,2,3,9,5,2,8,3,2,3,4,1,
%U 8,6,0,9,9,1,3,4,2,2,9,6,0,3,4,2,6,1,8,0,9,6,9,1,8,3,4,8,8,4,3,0
%N Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x).
%C For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*cos(x)=c*sin(x).
%C Guide to related sequences, with graphs included in Mathematica programs:
%C a.... b.... c.... x
%C 1.... 1.... 2.... A199597
%C 1.... 1.... 3.... A199598
%C 1.... 1.... 4.... A199599
%C 1.... 2.... 1.... A199600
%C 1.... 2.... 3.... A199601
%C 1.... 2.... 4.... A199602
%C 1.... 3.... 0.... A199603, A199604
%C 1.... 3.... 1.... A199605, A199606
%C 1.... 3.... 2.... A199607, A199608
%C 1.... 3.... 3.... A199609, A199610
%C 1.... 4.... 0.... A199611, A199612
%C 1.... 4.... 1.... A199613, A199614
%C 1.... 4.... 2.... A199615, A199616
%C 1.... 4.... 3.... A199617, A199618
%C 1.... 4.... 4.... A199619, A199620
%C 2.... 1.... 0.... A199621
%C 2.... 1.... 2.... A199622
%C 2.... 1.... 3.... A199623
%C 2.... 1.... 4.... A199624
%C 2.... 2.... 1.... A199625
%C 2.... 2.... 3.... A199661
%C 3.... 1.... 0.... A199662
%C 3.... 1.... 2.... A199663
%C 3.... 1.... 3.... A199664
%C 3.... 1.... 4.... A199665
%C 3.... 2.... 0.... A199666
%C 3.... 2.... 1.... A199667
%C 3.... 2.... 3.... A199668
%C 3.... 2.... 4.... A199669
%C 1... -1.... 0.... A003957
%C 1... -1.... 1.... A199722
%C 1... -1.... 2.... A199721
%C 1... -1.... 3.... A199720
%C 1... -1.... 4.... A199719
%C 1... -2.... 1.... A199726
%C 1... -2.... 2.... A199725
%C 1... -2.... 3.... A199724
%C 1... -2.... 4.... A199723
%C 1... -3.... 1.... A199730
%C 1... -3.... 2.... A199729
%C 1... -3.... 3.... A199728
%C 1... -3.... 4.... A199727
%C 1... -4.... 1.... A199737. A199738
%C 1... -4.... 2.... A199735, A199736
%C 1... -4.... 3.... A199733, A199734
%C 1... -4.... 4.... A199731. A199732
%C 2... -1.... 1.... A199742
%C 2... -1.... 2.... A199741
%C 2... -1.... 3.... A199740
%C 2... -1.... 4.... A199739
%C 2... -2.... 1.... A199776
%C 2... -2.... 3.... A199775
%C 2... -3.... 1.... A199780
%C 2... -3.... 2.... A199779
%C 2... -3.... 3.... A199778
%C 2... -3.... 4.... A199777
%C 2... -4.... 1.... A199782
%C 2... -4.... 3.... A199781
%C 3... -4.... 1.... A199786
%C 3... -4.... 2.... A199785
%C 3... -4.... 3.... A199784
%C 3... -4.... 4.... A199783
%C 3... -3.... 1.... A199789
%C 3... -3.... 2.... A199788
%C 3... -3.... 4.... A199787
%C 3... -2.... 1.... A199793
%C 3... -2.... 2.... A199792
%C 3... -2.... 3.... A199791
%C 3... -2.... 4.... A199790
%C 3... -1.... 1.... A199797
%C 3... -1.... 2.... A199796
%C 3... -1.... 3.... A199795
%C 3... -1.... 4.... A199794
%C 4... -4.... 1.... A199873
%C 4... -4.... 3.... A199872
%C 4... -3.... 1.... A199871
%C 4... -3.... 2.... A199870
%C 4... -3.... 3.... A199869
%C 4... -3.... 4.... A199868
%C 4... -2.... 1.... A199867
%C 4... -2.... 3.... A199866
%C 4... -1.... 1.... A199865
%C 4... -1.... 2.... A199864
%C 4... -1.... 3.... A199863
%C 4... -1.... 4.... A199862
%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 A199597, take f(x,u,v)=x^2+u*x*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.
%e 1.1881851344514388032178109729076525973...
%t (* Program 1: A199597 *)
%t a = 1; b = 1; c = 2;
%t f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
%t Plot[{f[x], g[x]}, {x, -Pi, Pi}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, 1.18, 1.19}, WorkingPrecision -> 110]
%t RealDigits[r] (* A199597 *)
%t (* Program 2: impl. surf. x^2+u*x*cos(x)=v*sin(x) *)
%t f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v*Sin[x];
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .5, 3}]}, {u, 0, 2}, {v, u, 20}];
%t ListPlot3D[Flatten[t, 1]] (* for A199597 *)
%Y Cf. A199370, A199170, A198866, A198755, A198414, A197737, A199429.
%K nonn,cons
%O 1,3
%A _Clark Kimberling_, Nov 08 2011
%E Edited by _Georg Fischer_, Aug 03 2021
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