A233592 Positive integers k such that the continued fraction expansion sqrt(k) = c(1) + c(1)/(c(2) + c(2)/(c(3) + c(3)/...)) is periodic.

2, 3, 5, 6, 8, 10, 11, 12, 15, 17, 18, 20, 24, 26, 27, 30, 35, 37, 38, 39, 40, 42, 44, 45, 48, 50, 51, 56, 63, 65, 66, 68, 72, 80, 82, 83, 84, 87, 90, 99, 101, 102, 104, 105, 108, 110, 120, 122, 123, 132, 143, 145

Offset: 1

Author

Stanislav Sykora, Jan 06 2014

Keywords

• nonn

Comments

For more details on this type of expansion, see A233582.

The cases with aperiodic expansions are listed in A233593.

All the listed cases become periodic after just two leading terms (it is a conjecture that this behavior is general); the validity of their expansions was explicitly tested.

Examples

			Blazys's expansion of sqrt(2) is {1, 2, 4, 4, 4, 4, 4, ...}, i.e., it has a periodic termination. Consequently, 2 is a term of this sequence.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..200
• Stanislav Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013.
• Stanislav Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki.

Crossrefs

Cf. A233593.

Cf. Blazys's expansions: A233582, A233584, A233585, A233586, A233587.

Programs

PARI

See the link.