A200684 -id:A200684 - cp's OEIS frontend

A200614 Decimal expansion of the lesser of two values of x satisfying 3*x^2 - 1 = tan(x) and 0 < x < Pi/2.

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

Offset: 0

Clark Kimberling, Nov 20 2011

For many choices of a and c, there are exactly two values of x satisfying a*x^2 - c = tan(x) and 0 < x < Pi/2; for other choices, there is exactly once such value.
Guide to related sequences, with graphs included in Mathematica programs:
a.... c.... x
3.... 1.... A200614, A200615
4.... 1.... A200616, A200617
5.... 1.... A200620, A200621
5.... 2.... A200622, A200623
5.... 3.... A200624, A200625
5.... 4.... A200626, A200627
5... -1.... A200628
5... -2.... A200629
5... -3.... A200630
5... -4.... A200631
6.... 1.... A200633, A200634
6.... 5.... A200635, A200636
6... -1.... A200637
6... -5.... A200638
1... -5.... A200639
2... -5.... A200640
3... -5.... A200641
4... -5.... A200642
2.... 0.... A200679, A200680
3.... 0.... A200681, A200682
4.... 0.... A200683, A200684
5.... 0.... A200618
6.... 0.... A200632
7.... 0.... A200643
8.... 0.... A200644
9.... 0.... A200645
10... 0.... A200646
-1... 1.... A200685
-1... 2.... A200686
-1... 3.... A200687
-1... 4.... A200688
-1... 5.... A200689
-1... 6.... A200690
-1... 7.... A200691
-1... 8.... A200692
-1... 9.... A200693
-1... 10... A200694
-2... 1.... A200695
-2... 3.... A200696
-3... 1.... A200697
-3... 2.... A200698
-4... 1.... A200699
-5... 1.... A200700
-6... 1.... A200701
-7... 1.... A200702
-8... 1.... A200703
-9... 1.... A200704
-10.. 1.... A200705
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 A200614, take f(x,u,v) = u*x^2 - v = tan(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.

			lesser:  0.839582259045302941513764008863804986308...
greater: 1.350956593976519397744696294368524715373...

Index entries for transcendental numbers.

Cf. A200615, A200338.

(* Program 1:  A200614 and A200615 *)
a = 3; c = 1;
f[x_] := a*x^2 - c; g[x_] := Tan[x]
Plot[{f[x], g[x]}, {x, -.1, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .8, .9}, WorkingPrecision -> 110]
RealDigits[r]   (* A200614 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.3, 1.4}, WorkingPrecision -> 110]
RealDigits[r]   (* A200615 *)
(* Program 2: implicit surface of u*x^2-v=tan(x) *)
f[{x_, u_, v_}] := u*x^2 - v - Tan[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1.55}]}, {u, 1, 20}, {v, -20, 0}];
ListPlot3D[Flatten[t, 1]]  (* for A200614 *)

A200683 Decimal expansion of the lesser of two values of x satisfying 4*x^2 = tan(x) and 0 < x < Pi/2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Nov 20 2011

See A200614 for a guide to related sequences. The Mathematica program includes a graph.

			lesser:  0.2555899667465678034714126335398146...
greater: 1.4529161609165145187427486875904483...

Index entries for transcendental numbers.

Cf. A200614.

a = 4; c = 0;
f[x_] := a*x^2 - c; g[x_] := Tan[x]
Plot[{f[x], g[x]}, {x, -.1, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .2, .3}, WorkingPrecision -> 110]
RealDigits[r](* A200683 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r](* A200684 *)

cp's OEIS Frontend

A200614 Decimal expansion of the lesser of two values of x satisfying 3*x^2 - 1 = tan(x) and 0 < x < Pi/2.

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