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%I #19 Aug 22 2018 08:30:31
%S 4,6,7,5,8,0,9,4,4,0,6,3,4,7,1,3,6,7,3,6,1,4,1,9,2,7,0,7,6,6,8,6,5,3,
%T 8,8,5,9,4,0,2,5,3,7,2,6,6,9,2,4,9,0,6,6,7,9,2,9,5,5,6,8,3,7,6,1,2,1,
%U 9,5,2,4,9,1,3,8,9,8,3,8,0,4,3,4,5,9,4,1,1,8,5,8,8,3,2,8,8,2,4
%N Decimal expansion of the least x satisfying x^2 + 2 = csc(x) and 0 < x < Pi.
%C For many choices of a and c, there are exactly two values of x satisfying a*x^2 + c = csc(x) and 0 < x < Pi. Guide to related sequences, with graphs included in Mathematica programs:
%C a.... c.... x
%C 1.... 1.... A196725, A201563
%C 1.... 2.... A201564, A201565
%C 1.... 3.... A201566, A201567
%C 1.... 4.... A201568, A201569
%C 1.... 5.... A201570, A201571
%C 1.... 6.... A201572, A201573
%C 1.... 7.... A201574, A201575
%C 1.... 8.... A201576, A201577
%C 1.... 9.... A201579, A201580
%C 1.... 10... A201578, A201581
%C 1.... 0.... A196617, A201582
%C 2.... 0.... A201583, A201584
%C 3.... 0.... A201585, A201586
%C 4.... 0.... A201587, A201588
%C 5.... 0.... A201589, A201590
%C 6.... 0.... A201591, A201653
%C 7.... 0.... A201654, A201655
%C 8.... 0.... A201656, A201657
%C 9.... 0.... A201658, A201659
%C 10... 0.... A201660, A201662
%C 1... -1.... A201661, A201663
%C 2... -1.... A201664, A201665
%C 3... -1.... A201666, A201667
%C 4... -1.... A201668, A201669
%C 5... -1.... A201670, A201671
%C 6... -1.... A201672, A201673
%C 7... -1.... A201674, A201675
%C 8... -1.... A201676, A201677
%C 9... -1.... A201678, A201679
%C 10.. -1.... A201680, A201681
%C 1... -2.... A201682, A201683
%C 1... -3.... A201735, A201736
%C 1... -4.... A201737, A201738
%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 A201564, take f(x,u,v)=u*x^2+v-csc(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.
%H G. C. Greubel, <a href="/A201564/b201564.txt">Table of n, a(n) for n = 0..10000</a>
%e least: 0.4675809440634713673614192707668653885...
%e greatest: 3.0531517225248702118041550531781137...
%t (* Program 1: A201564, A201565 *)
%t a = 1; c = 2;
%t f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, .46, .47}, WorkingPrecision -> 110]
%t RealDigits[r] (* A201564 *)
%t r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t RealDigits[r] (* A201565 *)
%t (* Program 2: implicit surface of u*x^2+v=csc(x) *)
%t f[{x_, u_, v_}] := u*x^2 + v - Csc[x];
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, 1}]}, {v, 0, 1}, {u, 2 + v, 10}];
%t ListPlot3D[Flatten[t, 1]] (* for A201564 *)
%o (PARI) a=1; c=2; solve(x=0.4, 0.5, a*x^2 + c - 1/sin(x)) // _G. C. Greubel_, Aug 21 2018
%Y Cf. A201397, A201565.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Dec 03 2011
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