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

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

1,3

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Mathematica

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

Crossrefs

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

Sequence in context: A350820 A371886 A280245 * A220963 A375566 A179313

Adjacent sequences: A290010 A290011 A290012 * A290014 A290015 A290016

Keyword

nonn,look,easy

Author

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

Status

approved