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 1-5 of 5 results.

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

A196619 Decimal expansion of the number c for which the curve y=cos(x) is tangent to the curve y=(1/x)-c, and 0

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 05 2011

Keywords

Examples

			x = 0.454451866354226599819691146329523402836346961...

Links

Crossrefs

Cf. A196617, A196618, A196620, A196612.

Programs

  • Mathematica
    Plot[{1/x - .4544, Cos[x]}, {x, 0, 2 Pi}]
xt = x /. FindRoot[x^(-2) == Sin[x], {x, .5, .8}, WorkingPrecision -> 100]
RealDigits[xt]      (* A196617 *)
Cos[xt]
RealDigits[Cos[xt]] (* A196618 *)
c = N[1/xt - Cos[xt], 100]
RealDigits[c]       (* A196619 *)
slope = -Sin[xt]
RealDigits[slope]   (* A196620 *)
  • PARI
    a=1; c=0; x=solve(x=1, 1.5, a*x^2 + c - 1/sin(x)); 1/x - cos(x) \\ G. C. Greubel, Aug 22 2018

A196618 Decimal expansion of cos(x), where x is the least positive solution of 1 = (x^2)*cos(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 05 2011

Keywords

Examples

			x = 0.4816817785482382699874297227751696380614905...

Links

Crossrefs

Cf. A196617, A196619.

Programs

  • Mathematica
    Plot[{1/x - .4544, Cos[x]}, {x, 0, 2 Pi}]
xt = x /. FindRoot[x^(-2) == Sin[x], {x, .5, .8}, WorkingPrecision -> 100]
RealDigits[xt]      (* A196617 *)
Cos[xt]
RealDigits[Cos[xt]] (* A196618 *)
c = N[1/xt - Cos[xt], 100]
RealDigits[c]       (* A196619 *)
slope = -Sin[xt]
RealDigits[slope]   (* A196620 *)
  • PARI
    a=1; c=0; x=solve(x=1, 1.5, a*x^2 + c - 1/sin(x)); cos(x) \\ G. C. Greubel, Aug 22 2018

Extensions

Terms a(85) onward corrected by G. C. Greubel, Aug 22 2018

A196620 Decimal expansion of the slope (negative) of the tangent line at the point of tangency of the curves y=cos(x) and y=(1/x)-c, where c is given by A196619.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 05 2011

Keywords

Examples

			x = -0.87634620111837419112349411392283024821317...

Links

Crossrefs

Cf. A196619.

Programs

  • Mathematica
    Plot[{1/x - .4544, Cos[x]}, {x, 0, 2 Pi}]
xt = x /. FindRoot[x^(-2) == Sin[x], {x, .5, .8}, WorkingPrecision -> 100]
RealDigits[xt]      (* A196617 *)
Cos[xt]
RealDigits[Cos[xt]] (* A196618 *)
c = N[1/xt - Cos[xt], 100]
RealDigits[c]       (* A196619 *)
slope = -Sin[xt]
RealDigits[slope]   (* A196620 *)
  • PARI
    a=1; c=0; x=solve(x=1, 1.5, a*x^2 + c - 1/sin(x)); -sin(x) \\ G. C. Greubel, Aug 22 2018

Extensions

Terms a(86) onward corrected by G. C. Greubel, Aug 22 2018

A201582 Decimal expansion of greatest x satisfying x^2 = csc(x) and 0

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 03 2011

Keywords

Comments

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

Examples

			least:  1.068223544197249018283471114263092898468...
greatest:  3.032645418388756188675325636802608932...

Links

Crossrefs

Cf. A201564, A196617.

Programs

  • Mathematica
    a = 1; c = 0;
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, .1, .2}, WorkingPrecision -> 110]
RealDigits[r]   (* A196617 *)
r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
RealDigits[r]   (* A201582 *)
  • PARI
    a=1; c=0; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ G. C. Greubel, Aug 22 2018
Showing 1-5 of 5 results.

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