A201564 Decimal expansion of the least x satisfying x^2 + 2 = csc(x) and 0 < x < Pi.

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

G. C. Greubel, Table of n, a(n) for n = 0..10000

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

Cf. A201397, A201565.

Sequence in context: A244816 A197585 A019605 * A200303 A309606 A288179

Adjacent sequences: A201561 A201562 A201563 * A201565 A201566 A201567

Keyword

nonn,cons

Author

Clark Kimberling, Dec 03 2011

Status

approved