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 A199597 Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x). 99
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*cos(x)=c*sin(x). Guide to related sequences, with graphs included in Mathematica programs: a.... b.... c.... x 1.... 1.... 2.... A199597 1.... 1.... 3.... A199598 1.... 1.... 4.... A199599 1.... 2.... 1.... A199600 1.... 2.... 3.... A199601 1.... 2.... 4.... A199602 1.... 3.... 0.... A199603, A199604 1.... 3.... 1.... A199605, A199606 1.... 3.... 2.... A199607, A199608 1.... 3.... 3.... A199609, A199610 1.... 4.... 0.... A199611, A199612 1.... 4.... 1.... A199613, A199614 1.... 4.... 2.... A199615, A199616 1.... 4.... 3.... A199617, A199618 1.... 4.... 4.... A199619, A199620 2.... 1.... 0.... A199621 2.... 1.... 2.... A199622 2.... 1.... 3.... A199623 2.... 1.... 4.... A199624 2.... 2.... 1.... A199625 2.... 2.... 3.... A199661 3.... 1.... 0.... A199662 3.... 1.... 2.... A199663 3.... 1.... 3.... A199664 3.... 1.... 4.... A199665 3.... 2.... 0.... A199666 3.... 2.... 1.... A199667 3.... 2.... 3.... A199668 3.... 2.... 4.... A199669 1... -1.... 0.... A003957 1... -1.... 1.... A199722 1... -1.... 2.... A199721 1... -1.... 3.... A199720 1... -1.... 4.... A199719 1... -2.... 1.... A199726 1... -2.... 2.... A199725 1... -2.... 3.... A199724 1... -2.... 4.... A199723 1... -3.... 1.... A199730 1... -3.... 2.... A199729 1... -3.... 3.... A199728 1... -3.... 4.... A199727 1... -4.... 1.... A199737. A199738 1... -4.... 2.... A199735, A199736 1... -4.... 3.... A199733, A199734 1... -4.... 4.... A199731. A199732 2... -1.... 1.... A199742 2... -1.... 2.... A199741 2... -1.... 3.... A199740 2... -1.... 4.... A199739 2... -2.... 1.... A199776 2... -2.... 3.... A199775 2... -3.... 1.... A199780 2... -3.... 2.... A199779 2... -3.... 3.... A199778 2... -3.... 4.... A199777 2... -4.... 1.... A199782 2... -4.... 3.... A199781 3... -4.... 1.... A199786 3... -4.... 2.... A199785 3... -4.... 3.... A199784 3... -4.... 4.... A199783 3... -3.... 1.... A199789 3... -3.... 2.... A199788 3... -3.... 4.... A199787 3... -2.... 1.... A199793 3... -2.... 2.... A199792 3... -2.... 3.... A199791 3... -2.... 4.... A199790 3... -1.... 1.... A199797 3... -1.... 2.... A199796 3... -1.... 3.... A199795 3... -1.... 4.... A199794 4... -4.... 1.... A199873 4... -4.... 3.... A199872 4... -3.... 1.... A199871 4... -3.... 2.... A199870 4... -3.... 3.... A199869 4... -3.... 4.... A199868 4... -2.... 1.... A199867 4... -2.... 3.... A199866 4... -1.... 1.... A199865 4... -1.... 2.... A199864 4... -1.... 3.... A199863 4... -1.... 4.... A199862 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 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. LINKS Table of n, a(n) for n=1..100. EXAMPLE 1.1881851344514388032178109729076525973... MATHEMATICA (* Program 1: A199597 *) a = 1; b = 1; c = 2; f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x] Plot[{f[x], g[x]}, {x, -Pi, Pi}, {AxesOrigin -> {0, 0}}] r = x /. FindRoot[f[x] == g[x], {x, 1.18, 1.19}, WorkingPrecision -> 110] RealDigits[r] (* A199597 *) (* Program 2: impl. surf. x^2+u*x*cos(x)=v*sin(x) *) f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v*Sin[x]; t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .5, 3}]}, {u, 0, 2}, {v, u, 20}]; ListPlot3D[Flatten[t, 1]] (* for A199597 *) CROSSREFS Cf. A199370, A199170, A198866, A198755, A198414, A197737, A199429. Sequence in context: A375153 A319858 A351210 * A366149 A197848 A224875 Adjacent sequences: A199594 A199595 A199596 * A199598 A199599 A199600 KEYWORD nonn,cons AUTHOR Clark Kimberling, Nov 08 2011 EXTENSIONS Edited by Georg Fischer, Aug 03 2021 STATUS approved

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Last modified August 12 10:46 EDT 2024. Contains 375092 sequences. (Running on oeis4.)