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 1-7 of 7 results.

A196766 Decimal expansion of the slope (negative) at the point of tangency of the curves y=c/x and y=sin(x), where c is given by A196765.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			x=-0.44212059295499839133561624405047613618690708...

Crossrefs

Cf. A196765.

Programs

  • Mathematica
    Plot[{Sin[x], 1/x, 1.82/x}, {x, 0, Pi}]
xt = x /. FindRoot[x + Tan[x] == 0, {x, 1.5, 2.5}, WorkingPrecision -> 100]
RealDigits[xt]   (* A196504 *)
c = N[xt*Sin[xt], 100]
RealDigits[c]    (* A196765 *)
slope = Cos[xt]
RealDigits[slope](* A196766 *)

A196760 Decimal expansion of the least x>0 satisfying 2=x*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			x=6.591467807276450408689193536456477366606...

Crossrefs

Cf. A196765.

Programs

  • Mathematica
    Plot[{1/x, 2/x, 3/x, 4/x, Sin[x]}, {x, 0, 4 Pi}]
t = x /. FindRoot[1/x == Sin[x], {x, 1, 1.2}, WorkingPrecision -> 100]
RealDigits[t]  (* A133866 *)
t = x /. FindRoot[2/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196760 *)
t = x /. FindRoot[3/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196761 *)
t = x /. FindRoot[4/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196762 *)
t = x /. FindRoot[5/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196763 *)
t = x /. FindRoot[6/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196764 *)

A196761 Decimal expansion of the least x>0 satisfying 3=x*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			x=6.7441683532591485558552811730153259440268799...

Crossrefs

Cf. A196765.

Programs

  • Mathematica
    Plot[{1/x, 2/x, 3/x, 4/x, Sin[x]}, {x, 0, 4 Pi}]
t = x /. FindRoot[1/x == Sin[x], {x, 1, 1.2}, WorkingPrecision -> 100]
RealDigits[t]  (* A133866 *)
t = x /. FindRoot[2/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196760 *)
t = x /. FindRoot[3/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196761 *)
t = x /. FindRoot[4/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196762 *)
t = x /. FindRoot[5/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196763 *)
t = x /. FindRoot[6/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196764 *)

A196762 Decimal expansion of the least x > 0 satisfying 4=x*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			6.9014126090213042932387323136399265516592011535132768777...

Crossrefs

Cf. A196765.

Programs

  • Mathematica
    t = x /. FindRoot[4/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]
  • PARI
    solve(x=6,7, 4/x-sin(x)) \\ Charles R Greathouse IV, Sep 06 2016

A196763 Decimal expansion of the least x>0 satisfying 5=x*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			x=7.068891403395066800687599561916762042715045101700...

Crossrefs

Cf. A196765.

Programs

  • Mathematica
    Plot[{1/x, 2/x, 3/x, 4/x, Sin[x]}, {x, 0, 4 Pi}]
t = x /. FindRoot[1/x == Sin[x], {x, 1, 1.2}, WorkingPrecision -> 100]
RealDigits[t]  (* A133866 *)
t = x /. FindRoot[2/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196760 *)
t = x /. FindRoot[3/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196761 *)
t = x /. FindRoot[4/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196762 *)
t = x /. FindRoot[5/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196763 *)
t = x /. FindRoot[6/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196764 *)

A196764 Decimal expansion of the least x>0 satisfying 6=x*sin(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			x=7.25663293662839986431355610086695712919471700...

Crossrefs

Cf. A196765.

Programs

  • Mathematica
    Plot[{1/x, 2/x, 3/x, 4/x, Sin[x]}, {x, 0, 4 Pi}]
t = x /. FindRoot[1/x == Sin[x], {x, 1, 1.2}, WorkingPrecision -> 100]
RealDigits[t]  (* A133866 *)
t = x /. FindRoot[2/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196760 *)
t = x /. FindRoot[3/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196761 *)
t = x /. FindRoot[4/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196762 *)
t = x /. FindRoot[5/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196763 *)
t = x /. FindRoot[6/x == Sin[x], {x, 6, 7}, WorkingPrecision -> 100]
RealDigits[t]   (* A196764 *)

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 1-7 of 7 results.

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