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 A197737 Decimal expansion of x<0 having x^2+x=cos(x). 144
 1, 2, 5, 1, 1, 5, 1, 8, 3, 5, 2, 2, 0, 7, 6, 4, 8, 1, 1, 5, 9, 2, 8, 7, 0, 0, 6, 8, 7, 8, 8, 1, 6, 1, 8, 5, 9, 9, 4, 5, 3, 5, 6, 1, 0, 8, 5, 8, 8, 9, 6, 8, 6, 3, 6, 2, 0, 1, 7, 8, 2, 8, 1, 2, 1, 0, 3, 6, 0, 1, 9, 1, 8, 2, 3, 8, 2, 1, 0, 9, 1, 0, 4, 1, 1, 2, 7, 3, 5, 7, 6, 5, 9, 4, 8, 6, 8, 4, 2 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For many choices of a,b,c, there are exactly two numbers x having a*x^2+b*x=cos(x). Guide to related sequences, with graphs included in Mathematica programs: a.... b.... c.... x 1.... 0.... 1.... A125578 1.... 0.... 2.... A197806 1.... 0.... 3.... A197807 1.... 0.... 4.... A197808 1.... 1.... 1.... A197737, A197738 1.... 1.... 2.... A197809, A197810 1.... 1.... 3.... A197811, A197812 1.... 1.... 4.... A197813, A197814 1... -2... -1.... A197815, A197820 1... -3... -1.... A197825, A197831 1... -4... -1.... A197839, A197840 1.... 2.... 1.... A197841, A197842 1.... 2.... 2.... A197843, A197844 1.... 2.... 3.... A197845, A197846 1.... 2.... 4.... A197847, A197848 1... -2... -2.... A197849, A197850 1... -3... -2.... A198098, A198099 1... -4... -2.... A198100, A198101 1.... 3.... 1.... A198102, A198103 1.... 3.... 2.... A198104, A198105 1.... 3.... 3.... A198106, A198107 1.... 3.... 4.... A198108, A198109 1... -2... -3.... A198140, A198141 1... -3... -3.... A198142, A198143 1... -4... -3.... A198144, A198145 2.... 0.... 1.... A198110 2.... 0.... 3.... A198111 2.... 1.... 1.... A198112, A198113 2.... 1.... 2.... A198114, A198115 2.... 1.... 3.... A198116, A198117 2.... 1.... 4.... A198118, A198119 2.... 1... -1.... A198120, A198121 2... -4... -1.... A198122, A198123 2.... 2.... 1.... A198124, A198125 2.... 2.... 3.... A198126, A198127 2.... 3.... 1.... A198128, A198129 2.... 3.... 2.... A198130, A198131 2.... 3.... 3.... A198132, A198133 2.... 3.... 4.... A198134, A198135 2... -4... -3.... A198136, A198137 3.... 0.... 1.... A198211 3.... 0.... 2.... A198212 3.... 0.... 4.... A198213 3.... 1.... 1.... A198214, A198215 3.... 1.... 2.... A198216, A198217 3.... 1.... 3.... A198218, A198219 3.... 1.... 4.... A198220, A198221 3.... 2.... 1.... A198222, A198223 3.... 2.... 2.... A198224, A198225 3.... 2.... 3.... A198226, A198227 3.... 2.... 4.... A198228, A198229 3.... 3.... 1.... A198230, A198231 3.... 3.... 2.... A198232, A198233 3.... 3.... 4.... A198234, A198235 3.... 4.... 1.... A198236, A198237 3.... 4.... 2.... A198238, A198239 3.... 4.... 3.... A198240, A198241 3.... 4.... 4.... A198138, A198139 3... -1... -1.... A198345, A198346 4.... 0.... 1.... A198347 4.... 0.... 3.... A198348 4.... 1.... 1.... A198349, A198350 4.... 1.... 2.... A198351, A198352 4.... 1.... 3.... A198353, A198354 4.... 1.... 4.... A198355, A198356 4.... 2.... 1.... A198357, A198358 4.... 2.... 3.... A198359, A198360 4.... 3.... 1.... A198361, A198362 4.... 3.... 2.... A198363, A198364 4.... 3.... 3.... A198365, A198366 4.... 3.... 4.... A198367, A198368 4.... 4.... 1.... A198369, A198370 4.... 4.... 3.... A198371, A198372 4... -4... -1.... A198373, A198374 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 A197737, take f(x,u,v)=x^2+u*x-v*cos(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 EXAMPLE negative: -1.25115183522076481159287006878816185994... positive:  0.55000934992726156666495361947172926116... MATHEMATICA (* Program 1:  A197738 *) a = 1; b = 1; c = 1; f[x_] := a*x^2 + b*x; g[x_] := c*Cos[x] Plot[{f[x], g[x]}, {x, -2, 1}] r1 = x /. FindRoot[f[x] == g[x], {x, -1.26, -1.25}, WorkingPrecision -> 110] RealDigits[r1] (* A197737 *) r1 = x /. FindRoot[f[x] == g[x], {x, .55, .551}, WorkingPrecision -> 110] RealDigits[r1] (* A197738 *) (* Program 2: implicit surface of x^2+u*x=v*cos(x) *) f[{x_, u_, v_}] := x^2 + u*x - v*Cos[x]; t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 20}, {v, u, 20}]; ListPlot3D[Flatten[t, 1]]  (* for A197737 *) CROSSREFS Cf. A197738. Sequence in context: A092134 A181779 A024548 * A189824 A197814 A091772 Adjacent sequences:  A197734 A197735 A197736 * A197738 A197739 A197740 KEYWORD nonn,cons AUTHOR Clark Kimberling, Oct 20 2011 STATUS approved

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Last modified September 20 01:52 EDT 2019. Contains 327207 sequences. (Running on oeis4.)