A117045 - OEIS

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A117045

Integers k (not perfect squares) such that the continued fraction expansion of the square root of k has period at most 2.

2, 3, 5, 6, 8, 10, 11, 15, 17, 18, 20, 24, 26, 27, 30, 35, 37, 38, 39, 40, 42, 48, 50, 51, 56, 63, 65, 66, 68, 72, 80, 82, 83, 84, 87, 90

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OFFSET

1,1

COMMENTS

In a recent paper, Justin Thomas, Julian Rosen, and I show that this is equivalent to the following criterion: let d be the integer part of the square root. Then sqrt(k) has period at most 2 if and only if 2d/(k - d^2) is an integer.

REFERENCES

Justin Thomas, Krishnan Shankar, Julian Rosen, "Continued Fractions, Square Roots and the orbit of 1/0 on the boundary of the hyperbolic plane", preprint.

LINKS

Table of n, a(n) for n=1..36.

K. Shankar, Square roots and continued fractions.

K. Shankar, SQUARE ROOTS, CONTINUED FRACTIONS AND THE ORBIT OF 1/0 ON dH2

EXAMPLE

The first term is 2 because sqrt(2) is irrational and for k=2, d=1, 2d/(k - d^2) = 1 is an integer.

CROSSREFS

Sequence in context: A179180 A085921 A005243 * A324793 A244053 A275833

Adjacent sequences: A117042 A117043 A117044 * A117046 A117047 A117048

KEYWORD

nonn

AUTHOR

Krishnan Shankar (shankar(AT)math.ou.edu), Apr 17 2006

STATUS

approved