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 1-3 of 3 results.

A196608 Decimal expansion of the least x>0 satisfying 1=x*cos(2*x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 05 2011

Keywords

Comments

Any solution other than 1 or 0 to an equation of the form x=t(f(x)) where t is a trigonometric function and f returns algebraic values for algebraic arguments is transcendental by the Lindemann-Weierstrass theorem. This means that all the solutions to the above equation as well as those in A196602, A196609 and A196626 are transcendental. - Chayim Lowen, Aug 15 2015

Examples

			x=2.55709109392790793745988777446340038675281809990...

Links

Crossrefs

Cf. A196602, A196609, A196626.

Programs

  • Mathematica
    Plot[{1/x, Cos[x], Cos[2 x], Cos[3 x], Cos[4 x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[1/x == Cos[x], {x, 4, 7}, WorkingPrecision -> 100]
RealDigits[t]  (* A133868 *)
t = x /. FindRoot[1/x == Cos[2 x], {x, 2, 3}, WorkingPrecision -> 100]
RealDigits[t]  (* A196608 *)
t = x /. FindRoot[1/x == Cos[3 x], {x, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196602 *)
t = x /. FindRoot[1/x == Cos[4 x], {x, .9, 1.4}, WorkingPrecision -> 100]
RealDigits[t]  (* A196609 *)
t = x /. FindRoot[1/x == Cos[5 x], {x, .9, 1.2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196626 *)

Formula

x is the limit of the iteration of x -> Pi - arccos(1/x)/2 on an initial argument a such that abs(a)>=1. - Chayim Lowen, Aug 16 2015

A196609 Decimal expansion of the least x>0 satisfying 1=x*cos(4*x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 05 2011

Keywords

Examples

			x=1.3806085256477567291281983692950566154588360255628744...

Programs

  • Mathematica
    Plot[{1/x, Cos[x], Cos[2 x], Cos[3 x], Cos[4 x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[1/x == Cos[x], {x, 4, 7}, WorkingPrecision -> 100]
RealDigits[t]  (* A133868 *)
t = x /. FindRoot[1/x == Cos[2 x], {x, 2, 3}, WorkingPrecision -> 100]
RealDigits[t]  (* A196608 *)
t = x /. FindRoot[1/x == Cos[3 x], {x, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196602 *)
t = x /. FindRoot[1/x == Cos[4 x], {x, .9, 1.4}, WorkingPrecision -> 100]
RealDigits[t]  (* A196609 *)
t = x /. FindRoot[1/x == Cos[5 x], {x, .9, 1.2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196626 *)

A196626 Decimal expansion of the least x>0 satisfying 1=x*cos(5*x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 05 2011

Keywords

Examples

			x=1.152561891218197606601460030599990...

Programs

  • Mathematica
    Plot[{1/x, Cos[x], Cos[2 x], Cos[3 x], Cos[4 x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[1/x == Cos[x], {x, 4, 7}, WorkingPrecision -> 100]
RealDigits[t]  (* A133868 *)
t = x /. FindRoot[1/x == Cos[2 x], {x, 2, 3}, WorkingPrecision -> 100]
RealDigits[t]  (* A196608 *)
t = x /. FindRoot[1/x == Cos[3 x], {x, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196602 *)
t = x /. FindRoot[1/x == Cos[4 x], {x, .9, 1.4}, WorkingPrecision -> 100]
RealDigits[t]  (* A196609 *)
t = x /. FindRoot[1/x == Cos[5 x], {x, .9, 1.2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196626 *)
Showing 1-3 of 3 results.

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