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Periods, n, of square root continued fractions at which A013646 increases.
3

%I #18 Oct 20 2019 01:59:24

%S 0,1,2,3,7,9,13,19,23,27,35,41,43,45,53,55,71,77,101,127,129,135,147,

%T 163,169,189,199,201,247,283,335,353,367,459,465,503,537,587,625,637,

%U 643,739,767,827,1009,1135,1325,1423,1433,1543,1561

%N Periods, n, of square root continued fractions at which A013646 increases.

%C Each term of this sequence takes a turn at being the smallest unknown period for a square root continued fraction. Periods 1 and 2 are seen as the periods of sqrt(2) and sqrt(3) respectively, but a period of 3 is not seen until sqrt(41).

%C By convention, the period for perfect squares (e.g., 1) is 0.

%C Open question: Are there any more even terms after the 2?

%H Patrick McKinley, <a href="/A215485/b215485.txt">Table of n, a(n) for n = 0..254</a>

%e When a square root continued fraction with a period of 3 is first seen (at sqrt(41)), the lowest period not yet seen is 7, which first occurs as the period of sqrt(58).

%Y Cf. A013646.

%K nonn

%O 0,3

%A _Patrick McKinley_, Aug 12 2012