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.

A196603 Decimal expansion of the least x>0 satisfying sec(x)=2x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 04 2011

Keywords

Examples

			0.61003128446417597537096307351341032...

Links

Crossrefs

Cf. A196610.

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 *)

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 *)

A295255 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*cos(x))).

Original entry on oeis.org

1, 1, 4, 27, 288, 4145, 75360, 1655003, 42628096, 1260274689, 42070233600, 1565308844539, 64237925148672, 2882670856605553, 140430196702035968, 7380867094885024635, 416320345406371921920, 25084955259883686000257, 1608058868442709001895936, 109278344982307590211482971
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A295237, A295254, A295256, A295257, A295258.

Programs

  • Maple
    a:=series(2/(1+sqrt(1-4*x*cos(x))),x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Cos[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Cos[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - x*cos(x)/(1 - x*cos(x)/(1 - x*cos(x)/(1 - x*cos(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2 - 2*r*sqrt(16*r^2 - 1)) * n^(n-1) / (exp(n) * r^n), where r = A196605 = 0.2585985822541894903... is the root of the equation r*cos(r) = 1/4. - Vaclav Kotesovec, Nov 18 2017

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
Showing 1-5 of 5 results.

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