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.

Previous Showing 31-33 of 33 results.

A290013 Length of the period of the continued fraction expansion of phi/n where phi is the golden ratio.

Original entry on oeis.org

1, 1, 2, 2, 1, 6, 2, 2, 6, 5, 4, 4, 1, 10, 8, 4, 3, 2, 8, 14, 2, 12, 10, 4, 11, 5, 14, 10, 4, 28, 8, 8, 8, 1, 20, 2, 7, 4, 8, 14, 6, 6, 18, 8, 24, 6, 2, 4, 22, 31, 12, 14, 9, 10, 2, 12, 16, 12, 20, 20, 5, 8, 8, 20, 13, 20, 22, 2, 10, 52, 28, 2, 15, 19, 36, 4
Offset: 1

Views

Author

Vanessa Gomez, Eric Jovinelly, Jacob A. McCann, Bryce Orloski, Catherine Rea, Shannon Talbott, Jul 17 2017

Keywords

Comments

We calculated the continued fraction expansion of phi/n and observed that the expansion is periodic after the first nonzero term. We tracked the periodicity of the expansions and present them here. The authors acknowledge the National Science Foundation (DMS-1560019) and Muhlenberg College for supporting the REU (Research Experiences for Undergraduates) on which this sequence is based.

Links

Crossrefs

Cf. A001622 (phi), A019863 (phi/2), A134943 (phi/3), A134944 (phi/4), A134946 (phi/6).

Programs

  • Mathematica
    a[n_] := ContinuedFraction[GoldenRatio/n] // Last // Length; Array[a, 80] (* Jean-François Alcover, Jul 28 2017 *)

A237128 Angles n expressed in degrees such that 2*cos(n) = phi where phi is the golden ratio (A001622).

Original entry on oeis.org

36, 324, 396, 684, 756, 1044, 1116, 1404, 1476, 1764, 1836, 2124, 2196, 2484, 2556, 2844, 2916, 3204, 3276, 3564, 3636, 3924, 3996, 4284, 4356, 4644, 4716, 5004, 5076, 5364, 5436, 5724, 5796, 6084, 6156, 6444, 6516, 6804, 6876, 7164, 7236, 7524, 7596, 7884
Offset: 1

Views

Author

Michel Lagneau, Feb 04 2014

Keywords

Comments

a(n) == 36, 324 mod 360 and a(n)/36 is congruent to {1,9} mod 10 (A090771).
See A019863 = half of the golden ratio (A001622) => a(1) = 90 - 54 degrees and a(2) = 360 - a(1) = 324 degrees.
The squares in the sequence are 36, 324, 1764, 2916, 4356, 6084, 10404, 12996, 15876, 19044, 26244, 30276, 34596, 39204, 49284, 54756, 60516, 66564, 79524,... with the following properties:
If a(n) == 36 mod 360 is a perfect square, sqrt(36+360*n)/6 = A090771 (numbers that are congruent to {1, 9} mod 10).
If a(n) == 324 mod 360 is a perfect square, sqrt(324+360*n)/6 = A063226 (numbers that are congruent to {3, 7} mod 10).

Examples

			1476 is in the sequence because 2*cos(1476°) = 2*cos(1476*Pi/180) = 1.61803398... = phi.

Links

Crossrefs

Cf. A001622, A019863.

Programs

  • Maple
    ***first program***
with(numtheory):err:=1/10^10:Digits:=20:for n from 1 to 20000 do:x:=evalf(2*cos(n*Pi/180)):ph:=evalf((1+sqrt(5)))/2:if abs(ph-x)
  • Mathematica
    Select[Range[8000],2*Cos[# Degree]==GoldenRatio&] (* or *) LinearRecurrence[ {1,1,-1},{36,324,396},50] (* Harvey P. Dale, Aug 14 2015 *)
  • PARI
    Vec(36*x*(x^2+8*x+1)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Feb 04 2014

Formula

a(n) = 18*(-5+3*(-1)^n+10*n). a(n) = a(n-1)+a(n-2)-a(n-3). G.f.: 36*x*(x^2+8*x+1) / ((x-1)^2*(x+1)). - Colin Barker, Feb 04 2014

A387445 Decimal expansion of cos(Pi/25).

Original entry on oeis.org

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

Views

Author

R. J. Mathar, Aug 29 2025

Keywords

Examples

			0.99211470131447783105...

Links

Crossrefs

Cf. A019694.

Formula

Equals sin(23*Pi/50).
Largest of the 10 real-valued roots of 1024*x^10 -2560*x^8 +2240*x^6 -32*x^5 -800*x^4 +40*x^3 +100*x^2 -10*x -1 =0.
16*this^5 -20*this^3 +5*this = A019863.
Previous Showing 31-33 of 33 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.