A308720 The maximum value in the continued fraction of sqrt(n), or 0 if there is no fractional part.

0, 0, 2, 2, 0, 4, 4, 4, 4, 0, 6, 6, 6, 6, 6, 6, 0, 8, 8, 8, 8, 8, 8, 8, 8, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 0, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 0, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 0, 16, 16, 16, 16, 16, 16, 16

Offset

0,3

Comments

The continued fraction expansion of sqrt(n) is periodic, and the maximal element is the last element in the period, 2*floor(sqrt(n)).

Links

Table of n, a(n) for n=0..71.

Oskar Perron, Die Lehre von den Kettenbrüchen, B. G. Teubner (1913), section 24, p. 87.

Formula

a(k^2) = 0.

a(m) = floor(sqrt(m)) for nonsquare m.

a(n) = 2 * A320471(n) for n > 0.

Mathematica

{0} ~Join~ Table[2 Mod[Floor@ Sqrt@ n, Ceiling@ Sqrt@ n], {n, 100}] (* Giovanni Resta, Jun 29 2019 *)

Crossrefs

Cf. A000196, A003285, A096494.

Sequence in context: A073469 A307076 A342472 * A086882 A341415 A328141

Adjacent sequences: A308717 A308718 A308719 * A308721 A308722 A308723

Keyword

nonn,easy

Author

Karl Fischer, Jun 19 2019

Extensions

More terms from Giovanni Resta, Jun 29 2019

Status

approved