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 1-9 of 9 results.

A196833 Decimal expansion of the slope (negative) at the point of tangency of the curves y=1/(1+x^2) and y=c*sin(x), where c is given by A196832.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			x=-0.12707184118644190594794464339300176838562544...

Crossrefs

Cf. A196825, A196832.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), .205 Sin[x]}, {x, 0, Pi}]
t = x /. FindRoot[x^2 + 2 x*Tan[x] + 1 == 0, {x, 2, 3}, WorkingPrecision -> 100]
RealDigits[t]     (* A196831 *)
c = N[Csc[t]/(1 + t^2), 100]
RealDigits[c]     (* A196832 *)
slope = N[c*Cos[t], 100]
RealDigits[slope] (* A196833 *)

A196825 Decimal expansion of the least x > 0 satisfying 1/(1 + x^2) = sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			0.7194212963274103157169229700373320490851010...

Links

Crossrefs

Cf. A196832.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Sin[x], 2 Sin[x], 3 Sin[x], 4 Sin[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196825 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196826 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196827 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196828 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196829 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196830 *)
  • PARI
    a=1; c=1; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ G. C. Greubel, Aug 21 2018

A196826 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=2*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			0.43420254999819638681352442196668401983962380...

Links

Crossrefs

Cf. A196832.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Sin[x], 2 Sin[x], 3 Sin[x], 4 Sin[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196825 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196826 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196827 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196828 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196829 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196830 *)

A196827 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=3*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			0.3091549335589725792525345241896404300813494203...

Links

Crossrefs

Cf. A196832.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Sin[x], 2 Sin[x], 3 Sin[x], 4 Sin[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196825 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196826 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196827 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196828 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196829 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196830 *)

A196828 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=4*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			0.238777659445904852564729030954613747638153989...

Links

Crossrefs

Cf. A196832.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Sin[x], 2 Sin[x], 3 Sin[x], 4 Sin[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196825 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196826 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196827 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196828 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196829 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196830 *)

A196829 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=5*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			0.1939624306810067166300804717739574865548853986...

Links

Crossrefs

Cf. A196832.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Sin[x], 2 Sin[x], 3 Sin[x], 4 Sin[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196825 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196826 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196827 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196828 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196829 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196830 *)

Extensions

Offset corrected by Georg Fischer, Aug 10 2021

A196830 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=6*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			0.16307121199550691891172025214962358231331887464030355...

Links

Crossrefs

Cf. A196832.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Sin[x], 2 Sin[x], 3 Sin[x], 4 Sin[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196825 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196826 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196827 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196828 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196829 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Sin[x], {x, 0, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196830 *)

A196831 Decimal expansion of the number x satisfying 0 < x < 2*Pi and x^2 + 2*x*tan(x) + 1 = 0.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			x=2.214416905079636330679565960367792215966376475...

Crossrefs

Cf. A196825, A196832.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), .205 Sin[x]}, {x, 0, Pi}]
t = x /. FindRoot[x^2 + 2 x*Tan[x] + 1 == 0, {x, 2, 3}, WorkingPrecision -> 100]
RealDigits[t]     (* A196831 *)
c = N[Csc[t]/(1 + t^2), 100]
RealDigits[c]     (* A196832 *)
slope = N[c*Cos[t], 100]
RealDigits[slope] (* A196833 *)

A196625 Decimal expansion of the number c for which the curve y=1/x is tangent to the curve y=cos(x-c), and 0 < x < 2*Pi; c = sqrt(r) - arccsc(r), where r = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 05 2011

Keywords

Comments

Let r=(1+sqrt(5))/2, the golden ratio. Let u=sqrt(r) and v=1/x. Let c=sqrt(r)-arccsc(r). The curve y=1/x is tangent to the curve y=cos(x-c) at (u,v), and the slope of the tangent line is r-1.
Guide to constants c associated with tangencies:
A196610: 1/x and c*cos(x)
A196619: 1/x - c and cos(x)
A196774: 1/x + c and sin(x)
A196625: 1/x and cos(c-x)
A196772: 1/x and sin(x+c)
A196758: 1/x and c*sin(x)
A196765: c/x and sin(x)
A196823: 1/(1+x^2) and -c+cos(x)
A196914: 1/(1+x^2) and c*cos(x)
A196832: 1/(1+x^2) and c*sin(x)
A197016: x=0, y=0, and cos(x)

Examples

			c=0.60578021702155370914841756575969877104...

Crossrefs

Cf. A139339, A196772.

Programs

  • Mathematica
    Plot[{1/x, Cos[x - 0.60578]}, {x, 0, 2 Pi}]
r = GoldenRatio; xt = Sqrt[r];
x1 = N[xt, 100]
RealDigits[x1]     (* A139339 *)
c = Sqrt[r] - ArcCsc[r];
c1 = N[c, 100]
RealDigits[c1]     (* A196625 *)
slope = N[r - Sqrt[5], 100]
RealDigits[slope]  (* -1+A001622; -1+golden ratio *)

Extensions

a(99) corrected by Georg Fischer, Jul 19 2021
Showing 1-9 of 9 results.

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