A200694 -id:A200694

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A200614

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

+10
61

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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.... A200239

2... -5.... A200240

3... -5.... A200241

4... -5.... A200242

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.

LINKS

Table of n, a(n) for n=0..98.

EXAMPLE

lesser: 0.839582259045302941513764008863804986308...

greater: 1.350956593976519397744696294368524715373...

MATHEMATICA

(* 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 *)

CROSSREFS

Cf. A200615, A200338.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Nov 20 2011

STATUS

approved

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