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A201564
Decimal expansion of the least x satisfying x^2 + 2 = csc(x) and 0 < x < Pi.
64
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, 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, 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
OFFSET
0,1
COMMENTS
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:
a.... c.... x
1.... 1.... A196725, A201563
1.... 2.... A201564, A201565
1.... 3.... A201566, A201567
1.... 4.... A201568, A201569
1.... 5.... A201570, A201571
1.... 6.... A201572, A201573
1.... 7.... A201574, A201575
1.... 8.... A201576, A201577
1.... 9.... A201579, A201580
1.... 10... A201578, A201581
1.... 0.... A196617, A201582
2.... 0.... A201583, A201584
3.... 0.... A201585, A201586
4.... 0.... A201587, A201588
5.... 0.... A201589, A201590
6.... 0.... A201591, A201653
7.... 0.... A201654, A201655
8.... 0.... A201656, A201657
9.... 0.... A201658, A201659
10... 0.... A201660, A201662
1... -1.... A201661, A201663
2... -1.... A201664, A201665
3... -1.... A201666, A201667
4... -1.... A201668, A201669
5... -1.... A201670, A201671
6... -1.... A201672, A201673
7... -1.... A201674, A201675
8... -1.... A201676, A201677
9... -1.... A201678, A201679
10.. -1.... A201680, A201681
1... -2.... A201682, A201683
1... -3.... A201735, A201736
1... -4.... A201737, A201738
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 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.
LINKS
EXAMPLE
least: 0.4675809440634713673614192707668653885...
greatest: 3.0531517225248702118041550531781137...
MATHEMATICA
(* Program 1: A201564, A201565 *)
a = 1; c = 2;
f[x_] := a*x^2 + c; g[x_] := Csc[x]
Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .46, .47}, WorkingPrecision -> 110]
RealDigits[r] (* A201564 *)
r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
RealDigits[r] (* A201565 *)
(* Program 2: implicit surface of u*x^2+v=csc(x) *)
f[{x_, u_, v_}] := u*x^2 + v - Csc[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, 1}]}, {v, 0, 1}, {u, 2 + v, 10}];
ListPlot3D[Flatten[t, 1]] (* for A201564 *)
PROG
(PARI) a=1; c=2; solve(x=0.4, 0.5, a*x^2 + c - 1/sin(x)) \\ G. C. Greubel, Aug 21 2018
CROSSREFS
Sequence in context: A244816 A197585 A019605 * A200303 A309606 A288179
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Dec 03 2011
STATUS
approved