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.

A196604 Decimal expansion of the least x>0 satisfying sec(x)=3x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 04 2011

Keywords

Examples

			0.3555759893429733726256531085657759489...

Links

Programs

  • Mathematica
    Plot[{1/x, Cos[x], 2 Cos[x], 3*Cos[x], 4 Cos[x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[1/x == Cos[x], {x, .1, 5}, WorkingPrecision -> 100]
RealDigits[t]  (* A133868 *)
t = x /. FindRoot[1/x == 2 Cos[x], {x, .5, .7}, WorkingPrecision -> 100]
RealDigits[t]  (* A196603 *)
t = x /. FindRoot[1/x == 3 Cos[x], {x, .3, .4}, WorkingPrecision -> 100]
RealDigits[t]  (* A196604 *)
t = x /. FindRoot[1/x == 4 Cos[x], {x, .1, .3}, WorkingPrecision -> 100]
RealDigits[t]  (* A196605 *)
t = x /. FindRoot[1/x == 5 Cos[x], {x, .15, .23}, WorkingPrecision -> 100]
RealDigits[t]  (* A196606 *)
t = x /. FindRoot[1/x == 6 Cos[x], {x, .1, .2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196607 *)

A196605 Decimal expansion of the least x>0 satisfying sec(x)=4x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 04 2011

Keywords

Examples

			0.25859858225418949030448261956152028133855...

Links

Crossrefs

Cf. A295255.

Programs

  • Mathematica
    Plot[{1/x, Cos[x], 2 Cos[x], 3*Cos[x], 4 Cos[x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[1/x == Cos[x], {x, .1, 5}, WorkingPrecision -> 100]
RealDigits[t]  (* A133868 *)
t = x /. FindRoot[1/x == 2 Cos[x], {x, .5, .7}, WorkingPrecision -> 100]
RealDigits[t]  (* A196603 *)
t = x /. FindRoot[1/x == 3 Cos[x], {x, .3, .4}, WorkingPrecision -> 100]
RealDigits[t]  (* A196604 *)
t = x /. FindRoot[1/x == 4 Cos[x], {x, .1, .3}, WorkingPrecision -> 100]
RealDigits[t]  (* A196605 *)
t = x /. FindRoot[1/x == 5 Cos[x], {x, .15, .23}, WorkingPrecision -> 100]
RealDigits[t]  (* A196606 *)
t = x /. FindRoot[1/x == 6 Cos[x], {x, .1, .2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196607 *)

A196606 Decimal expansion of the least x>0 satisfying sec(x)=5x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 04 2011

Keywords

Examples

			0.2042453787045389017234590570552809773445731130635969...

Links

Programs

  • Mathematica
    Plot[{1/x, Cos[x], 2 Cos[x], 3*Cos[x], 4 Cos[x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[1/x == Cos[x], {x, .1, 5}, WorkingPrecision -> 100]
RealDigits[t]  (* A133868 *)
t = x /. FindRoot[1/x == 2 Cos[x], {x, .5, .7}, WorkingPrecision -> 100]
RealDigits[t]  (* A196603 *)
t = x /. FindRoot[1/x == 3 Cos[x], {x, .3, .4}, WorkingPrecision -> 100]
RealDigits[t]  (* A196604 *)
t = x /. FindRoot[1/x == 4 Cos[x], {x, .1, .3}, WorkingPrecision -> 100]
RealDigits[t]  (* A196605 *)
t = x /. FindRoot[1/x == 5 Cos[x], {x, .15, .23}, WorkingPrecision -> 100]
RealDigits[t]  (* A196606 *)
t = x /. FindRoot[1/x == 6 Cos[x], {x, .1, .2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196607 *)

A196607 Decimal expansion of the least x>0 satisfying sec(x)=6x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 04 2011

Keywords

Examples

			0.169077647398015149995295367672627810742134076969653717056...

Links

Programs

  • Mathematica
    Plot[{1/x, Cos[x], 2 Cos[x], 3*Cos[x], 4 Cos[x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[1/x == Cos[x], {x, .1, 5}, WorkingPrecision -> 100]
RealDigits[t]  (* A133868 *)
t = x /. FindRoot[1/x == 2 Cos[x], {x, .5, .7}, WorkingPrecision -> 100]
RealDigits[t]  (* A196603 *)
t = x /. FindRoot[1/x == 3 Cos[x], {x, .3, .4}, WorkingPrecision -> 100]
RealDigits[t]  (* A196604 *)
t = x /. FindRoot[1/x == 4 Cos[x], {x, .1, .3}, WorkingPrecision -> 100]
RealDigits[t]  (* A196605 *)
t = x /. FindRoot[1/x == 5 Cos[x], {x, .15, .23}, WorkingPrecision -> 100]
RealDigits[t]  (* A196606 *)
t = x /. FindRoot[1/x == 6 Cos[x], {x, .1, .2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196607 *)
  • PARI
    solve(x=0,1,6*x*cos(x)-1) \\ Charles R Greathouse IV, Aug 23 2021

A196610 Decimal expansion of the number c for which the curve y = c*cos(x) is tangent to the curve y = 1/x, and 0 < x < 2*Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 04 2011

Keywords

Examples

			c=1.7822251402031333124075510326661600195134426369...

Crossrefs

Cf. A196603, A196611.

Programs

  • Mathematica
    Plot[{1/x, (1.78222) Cos[x]}, {x, .7, 1}]
xt = x /. FindRoot[x == Cot[x], {x, .8, 1}, WorkingPrecision -> 100]
c = N[Csc[xt]/xt^2, 100]
RealDigits[c]      (* A196610 *)
slope = -c*Sin[xt]
RealDigits[slope]  (* A196611 *)

A352646 Expansion of e.g.f. 1/(1 - 2 * x * cos(x)).

Original entry on oeis.org

1, 2, 8, 42, 288, 2410, 24000, 277186, 3648512, 53936082, 885150720, 15970846298, 314273439744, 6698574264122, 153746319720448, 3780677636321010, 99163499845386240, 2763481838977368994, 81542013760903053312, 2539717324111483027594
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Links

Crossrefs

Cf. A352252, A352647.
Cf. A352642, A352648.
Cf. A185951.

Programs

  • Mathematica
    With[{m = 19}, Range[0, m]! * CoefficientList[Series[1/(1 - 2*x*Cos[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x*cos(x))))
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / ((1 - r * sqrt(4*r^2 - 1)) * r^n), where r = A196603 = 0.6100312844641759753709630735134103246737209791121692378637516075328... is the root of the equation 2*r*cos(r) = 1. - Vaclav Kotesovec, Mar 27 2022
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025

A196611 Decimal expansion of the slope (negative) of the tangent line at the point of tangency of the curves y=c*cos(x) and y=1/x, where c is given by A196610.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 04 2011

Keywords

Comments

For x>0, there is exactly one number c for which the graphs of y=c*cos(x) and y=1/x, where 0

Examples

			slope = -1.3510338868783786240091924735284302174...

Crossrefs

Cf. A196610, A196603.

Programs

  • Mathematica
    Plot[{1/x, (1.78222) Cos[x]}, {x, .7, 1}]
xt = x /. FindRoot[x == Cot[x], {x, .8, 1}, WorkingPrecision -> 100]
c = N[Csc[xt]/xt^2, 100]
RealDigits[c]  (* A196610 *)
slope = -c*Sin[xt]
RealDigits[slope]  (* A196611 *)
Showing 1-7 of 7 results.

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