A059853 - OEIS

A059853

Period of continued fraction for sqrt(n^2+3), n >= 2.

4, 2, 6, 4, 2, 6, 10, 2, 12, 16, 2, 16, 20, 2, 10, 10, 2, 12, 10, 2, 28, 10, 2, 26, 16, 2, 18, 48, 2, 34, 12, 2, 26, 32, 2, 32, 32, 2, 20, 70, 2, 56, 34, 2, 24, 22, 2, 54, 52, 2, 70, 16, 2, 18, 38, 2, 16, 36, 2, 12, 72, 2, 114, 30, 2, 64, 32, 2, 52, 54, 2, 22, 92, 2, 154, 88, 2, 56

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OFFSET

2,1

COMMENTS

The old name was "Quotient cycle length of sqrt(n^2+3)." - Jianing Song, May 01 2021

LINKS

Amiram Eldar, Table of n, a(n) for n = 2..10000

FORMULA

If n is a multiple of 3 then a(n) = 2.

a(n) = A003285(n^2+3). - Jianing Song, May 01 2021

EXAMPLE

sqrt(35^2+3) = [35; 23, 2, 1, 7, 8, 1, 1, 1, 2, 2, 1, 1, 5, 3, 1, 16, 1, 3, 5, 1, 1, 2, 2, 1, 1, 1, 8, 7, 1, 2, 23, 70], so a(35) = 32.

sqrt(36^2+3) = [36; 24, 72], so a(36) = 2.

sqrt(37^2+3) = [37; 24, 1, 2, 7, 1, 8, 2, 1, 1, 1, 2, 2, 5, 1, 3, 18, 3, 1, 5, 2, 2, 1, 1, 1, 2, 8, 1, 7, 2, 1, 24, 74], so a(37) = 32.

MAPLE

with(numtheory): [seq(nops(cfrac(sqrt(k^2+3), 'periodic', 'quotients')[2]), k=2..256)];

MATHEMATICA

a[n_] := Length[ContinuedFraction[Sqrt[n^2 + 3]][[2]]]; Array[a, 100, 2] (* Amiram Eldar, Jul 10 2024 *)

CROSSREFS

Cf. A003285.

Period of continued fraction for sqrt(n^2+k): this sequence (k=3), A059855 (k=4), A059854 (k=5).

Sequence in context: A242599 A092205 A272101 * A136527 A246776 A138614

Adjacent sequences: A059850 A059851 A059852 * A059854 A059855 A059856

KEYWORD

nonn

AUTHOR

Labos Elemer, Feb 27 2001

EXTENSIONS

New name by Jianing Song, May 01 2021

STATUS

approved