cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 45 results. Next

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 03 2011

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A201397, A201565.

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

A201406 Decimal expansion of least x satisfying 2*x^2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

See A201397 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.892874306058961124447371969018677514...
greatest: 1.2390826209275819187151092399019855...

Links

Crossrefs

Cf. A201397.

Programs

  • Mathematica
    a = 2; c = 0;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .8, 1}, WorkingPrecision -> 110]
RealDigits[r]   (* A201406 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]   (* A201407 *)

A201407 Decimal expansion of greatest x satisfying 2*x^2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

See A201397 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.892874306058961124447371969018677514...
greatest: 1.2390826209275819187151092399019855...

Links

Crossrefs

Cf. A201397.

Programs

  • Mathematica
    a = 2; c = 0;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .8, 1}, WorkingPrecision -> 110]
RealDigits[r]   (* A201406 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]   (* A201407 *)

A201408 Decimal expansion of least x satisfying 3*x^2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

See A201397 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.6461374540628972972901679159101125226952859...
greatest: 1.39986411944606406722963950518361037394178...

Links

Crossrefs

Cf. A201397.

Programs

  • Mathematica
    a = 3; c = 0;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .6, .7}, WorkingPrecision -> 110]
RealDigits[r]    (* A201408 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.3, 1.4}, WorkingPrecision -> 110]
RealDigits[r]    (* A201409 *)

A201409 Decimal expansion of greatest x satisfying 3*x^2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

See A201397 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.6461374540628972972901679159101125226952859...
greatest: 1.39986411944606406722963950518361037394178...

Links

Crossrefs

Cf. A201397.

Programs

  • Mathematica
    a = 3; c = 0;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .6, .7}, WorkingPrecision -> 110]
RealDigits[r]    (* A201408 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.3, 1.4}, WorkingPrecision -> 110]
RealDigits[r]    (* A201409 *)

A201410 Decimal expansion of least x satisfying 4*x^2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

See A201397 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.53986108391277844363067373273228071480624...
greatest: 1.451925722123287999446946604502079960054...

Links

Crossrefs

Cf. A201397.

Programs

  • Mathematica
    a = 4; c = 0;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
RealDigits[r]    (* A201410 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]    (* A201411 *)

A201411 Decimal expansion of greatest x satisfying 4*x^2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

See A201397 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.53986108391277844363067373273228071480624...
greatest: 1.451925722123287999446946604502079960054...

Links

Crossrefs

Cf. A201397.

Programs

  • Mathematica
    a = 4; c = 0;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
RealDigits[r]    (* A201410 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]    (* A201411 *)

A201412 Decimal expansion of least x satisfying 5*x^2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

See A201397 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.474127690420775415934748938569551538434...
greatest: 1.4792710652904107931042853415537602633...

Links

Crossrefs

Cf. A201397.

Programs

  • Mathematica
    a = 5; c = 0;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[r]     (* A201412 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]    (* A201413 *)

A201413 Decimal expansion of greatest x satisfying 5*x^2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

See A201397 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.474127690420775415934748938569551538434...
greatest: 1.4792710652904107931042853415537602633...

Links

Crossrefs

Cf. A201397.

Programs

  • Mathematica
    a = 5; c = 0;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[r]    (* A201412 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]    (* A201413 *)

A201414 Decimal expansion of least x satisfying 6*x^2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

See A201397 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.42800895010041097002739347769069180659...
greatest: 1.496285048607652953479229041712424469...

Links

Crossrefs

Cf. A201397.

Programs

  • Mathematica
    a = 6; c = 0;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[r]    (* A201414 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]    (* A201415 *)
Showing 1-10 of 45 results. Next

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