A200614 -id:A200614 - cp's OEIS frontend

A201280 Decimal expansion of x satisfying x^2 + 1 = cot(x) and 0 < x < Pi.

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

Offset: 0

Clark Kimberling, Nov 29 2011

For many choices of a and c, there is exactly one x satisfying a*x^2 + c = cot(x) and 0 < x < Pi.
Guide to related sequences, with graphs included in Mathematica programs:
a.... c.... x
... 1.... A201280
... 2.... A201281
... 3.... A201282
... 4.... A201283
... 5.... A201284
... 6.... A201285
... 7.... A201286
... 8.... A201287
... 9.... A201288
... 10... A201289
... 0.... A201294
.. -1.... A201295
.. -2.... A201296
.. -3.... A201297
.. -4.... A201298
.. -5.... A201299
.. -6.... A201315
.. -7.... A201316
.. -8.... A201317
.. -9.... A201318
. -10.... A201319
... 0.... A201329
... 0.... A201330
... 0.... A201331
... 0.... A201332
... 0.... A201333
... 0.... A201334
... 0.... A201335
... 0.... A201336
.. 0.... A201337
.. -1.... A201320
.. -1.... A201321
.. -1.... A201322
.. -1.... A201323
.. -1.... A201324
.. -1.... A201325
.. -1.... A201326
.. -1.... A201327
. -1.... A201328
... 1.... A201290
... 3.... A201291
.. -3.... A201394
... 1.... A201292
... 2.... A201293
.. -2.... A201395
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 A201280, take f(x,u,v) = u*x^2 - v - cot(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.

			0.62389956058090344363990329394632442...

Index entries for transcendental numbers.

Cf. A200614.

(* Program 1: A201280 *)
a = 1; c = 1;
f[x_] := a*x^2 + c; g[x_] := Cot[x]
Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .62, .63}, WorkingPrecision -> 110]
RealDigits[r]   (* A201280 *)
(* Program 2: implicit surface of u*x^2-v=cot(x) *)
f[{x_, u_, v_}] := u*x^2 - v - Cot[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .001, Pi}]}, {u, 0, 5, .1}, {v, 0, 5, .1}];
ListPlot3D[Flatten[t, 1]]  (* for A201280 *)

Edited and a(90) onwards corrected by Georg Fischer, Aug 03 2021

A201397 Decimal expansion of x satisfying x^2 + 2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 01 2011

For many choices of a and c, there are exactly two values of x satisfying a*x^2 + c = sec(x) and 0 < x < Pi. Guide to related sequences, with graphs included in Mathematica programs:
a.... c.... x
... 1.... A196816
... 2.... A201397
... 3.... A201398
... 4.... A201399
... 5.... A201400
... 6.... A201401
... 7.... A201402
... 8.... A201403
... 9.... A201404
... 10... A201405
... 0.... A201406, A201407
... 0.... A201408, A201409
... 0.... A201410, A201411
... 0.... A201412, A201413
... 0.... A201414, A201415
... 0.... A201416, A201417
... 0.... A201418, A201419
... 0.... A201420, A201421
.. 0.... A201422, A201423
.. -1.... A201515, A201516
.. -1.... A201517, A201518
.. -1.... A201519, A201520
.. -1.... A201521, A201522
.. -1.... A201523, A201524
.. -1.... A201525, A201526
.. -1.... A201527, A201528
. -1.... A201529, A201530
... 3.... A201531
... 2.... A200619
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 A201397, take f(x,u,v) = u*x^2 + v = sec(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.

			1.2954596464154787686299132707186415897672...

Index entries for transcendental numbers.

Cf. A201280, A200614.

(* Program 1:  A201397 *)
a = 1; c = 2;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]    (* A201397 *)
(* Program 2: implicit surface of u*x^2+v=sec(x) *)
Remove["Global`*"];
f[{x_, u_, v_}] := u*x^2 + v - Sec[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 A201397 *)

A200616 Decimal expansion of the lesser of two values of x satisfying 4*x^2 - 1 = tan(x) and 0 < x < Pi/2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Nov 20 2011

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

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

Index entries for transcendental numbers.

Cf. A200614.

a = 4; 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, .6, .7}, WorkingPrecision -> 110]
RealDigits[r]   (* A200616 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]   (* A200617 *)

A200617 Decimal expansion of the greater of two values of x satisfying 4*x^2 - 1 = tan(x) and 0 < x < Pi/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 20 2011

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

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

Index entries for transcendental numbers.

Cf. A200614, A200616.

a = 4; 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, .6, .7}, WorkingPrecision -> 110]
RealDigits[r]   (* A200616 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]   (* A200617 *)

A200620 Decimal expansion of the lesser of two values of x satisfying 5*x^2 - 1 = tan(x) and 0 < x < Pi/2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Nov 20 2011

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

			lesser:  0.5738256142207075194706993073950289720400...
greater: 1.469002719513610613223362597583632411278000...

Index entries for transcendental numbers.

Cf. A200614, A200621.

a = 5; 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, .5, .6}, WorkingPrecision -> 110]
RealDigits[r]   (* A200620 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]   (* A200621 *)

A200621 Decimal expansion of the greater of two values of x satisfying 5*x^2 - 1 = tan(x) and 0 < x < Pi/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 20 2011

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

			lesser:  0.5738256142207075194706993073950289720400...
greater: 1.469002719513610613223362597583632411278000...

Index entries for transcendental numbers.

Cf. A200614.

a = 5; 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, .5, .6}, WorkingPrecision -> 110]
RealDigits[r]   (* A200620 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]   (* A200621 *)

A200622 Decimal expansion of the lesser of two values of x satisfying 5*x^2 - 2 = tan(x) and 0 < x < Pi/2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Nov 20 2011

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

			lesser:  0.770896883914277182837264927358706321868754...
greater: 1.454799213519995526370784300798944589012608...

Index entries for transcendental numbers.

Cf. A200614.

a = 5; c = 2;
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, .7, .8}, WorkingPrecision -> 110]
RealDigits[r]   (* A200622 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]   (* A200623 *)

A200623 Decimal expansion of the greater of two values of x satisfying 5*x^2 - 2 = tan(x) and 0 < x < Pi/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 20 2011

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

			lesser:  0.770896883914277182837264927358706321868754...
greater: 1.4547992135199955263707843007989445890126087...

Index entries for transcendental numbers.

Cf. A200614.

a = 5; c = 2;
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, .7, .8}, WorkingPrecision -> 110]
RealDigits[r]    (* A200622 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]   (* A200623 *)

A200624 Decimal expansion of the lesser of two values of x satisfying 5*x^2 - 3 = tan(x) and 0 < x < Pi/2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Nov 20 2011

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

			lesser:  0.9325170518642294819498571898931399897649173...
greater: 1.43443679853106488271886435135433585034396681...

Index entries for transcendental numbers.

Cf. A200614.

a = 5; c = 3;
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, .93, .94}, WorkingPrecision -> 110]
RealDigits[r]    (* A200624 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]   (* A200625 *)

A200625 Decimal expansion of the greater of two values of x satisfying 5*x^2 - 3 = tan(x) and 0 < x < Pi/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 20 2011

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

			lesser:  0.9325170518642294819498571898931399897...
greater: 1.4344367985310648827188643513543358503...

Index entries for transcendental numbers.

Cf. A200614.

a = 5; c = 3;
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, .93, .94}, WorkingPrecision -> 110]
RealDigits[r]   (* A200624 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]   (* A200625 *)

cp's OEIS Frontend

A201280 Decimal expansion of x satisfying x^2 + 1 = cot(x) and 0 < x < Pi.

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A201397 Decimal expansion of x satisfying x^2 + 2 = sec(x) and 0 < x < Pi.

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A200616 Decimal expansion of the lesser of two values of x satisfying 4*x^2 - 1 = tan(x) and 0 < x < Pi/2.

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A200617 Decimal expansion of the greater of two values of x satisfying 4*x^2 - 1 = tan(x) and 0 < x < Pi/2.

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Mathematica

A200620 Decimal expansion of the lesser of two values of x satisfying 5*x^2 - 1 = tan(x) and 0 < x < Pi/2.

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Mathematica

A200621 Decimal expansion of the greater of two values of x satisfying 5*x^2 - 1 = tan(x) and 0 < x < Pi/2.

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Mathematica

A200622 Decimal expansion of the lesser of two values of x satisfying 5*x^2 - 2 = tan(x) and 0 < x < Pi/2.

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Mathematica

A200623 Decimal expansion of the greater of two values of x satisfying 5*x^2 - 2 = tan(x) and 0 < x < Pi/2.

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