A201565 Decimal expansion of greatest x satisfying x^2 + 2 = csc(x) and 0 < x < Pi.

3, 0, 5, 3, 1, 5, 1, 7, 2, 2, 5, 2, 4, 8, 7, 0, 2, 1, 1, 8, 0, 4, 1, 5, 5, 0, 5, 3, 1, 7, 8, 1, 1, 3, 7, 4, 5, 9, 6, 2, 2, 4, 7, 6, 7, 8, 3, 9, 2, 0, 5, 5, 3, 4, 7, 5, 4, 1, 5, 4, 4, 1, 3, 9, 0, 6, 3, 7, 7, 3, 7, 1, 6, 9, 0, 6, 9, 5, 2, 2, 2, 7, 9, 1, 6, 9, 7, 4, 3, 4, 0, 3, 5, 9, 3, 5, 7, 5, 5

Offset: 1

Author

Clark Kimberling, Dec 03 2011

Keywords

• nonn
• cons

Comments

See A201564 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.46758094406347136736141927076686538859402537...
greatest:  3.05315172252487021180415505317811374596224...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for transcendental numbers.

Crossrefs

Cf. A201564.

Programs

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 *)

PARI

a=1; c=2; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ G. C. Greubel, Aug 21 2018