A201670 -id:A201670 - cp's OEIS frontend

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

Clark Kimberling, Dec 03 2011

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.... A196825, A201563
... 2.... A201564, A201565
... 3.... A201566, A201567
... 4.... A201568, A201569
... 5.... A201570, A201571
... 6.... A201572, A201573
... 7.... A201574, A201575
... 8.... A201576, A201577
... 9.... A201579, A201580
... 10... A201578, A201581
... 0.... A196617, A201582
... 0.... A201583, A201584
... 0.... A201585, A201586
... 0.... A201587, A201588
... 0.... A201589, A201590
... 0.... A201591, A201653
... 0.... A201654, A201655
... 0.... A201656, A201657
... 0.... A201658, A201659
.. 0.... A201660, A201662
.. -1.... A201661, A201663
.. -1.... A201664, A201665
.. -1.... A201666, A201667
.. -1.... A201668, A201669
.. -1.... A201670, A201671
.. -1.... A201672, A201673
.. -1.... A201674, A201675
.. -1.... A201676, A201677
.. -1.... A201678, A201679
. -1.... A201680, A201681
.. -2.... A201682, A201683
.. -3.... A201735, A201736
.. -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.

			least:  0.4675809440634713673614192707668653885...
greatest:  3.0531517225248702118041550531781137...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A201397, A201565.

(* 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 *)

a=1; c=2; solve(x=0.4, 0.5, a*x^2 + c - 1/sin(x)) \\ G. C. Greubel, Aug 21 2018

A201671 Decimal expansion of greatest x satisfying 5*x^2 - 1 = csc(x) and 0

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 04 2011

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

			least:  0.711533888367151499246796307428212556772...
greatest:  3.120622993086918805141273424623066002...

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

Cf. A201564.

a = 5; c = -1;
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, .7, .8}, WorkingPrecision -> 110]
RealDigits[r]     (* A201670 *)
r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
RealDigits[r]     (* A201671 *)

a=5; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ G. C. Greubel, Sep 11 2018

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

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