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Patterns in Continued Fractions of Square Roots
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Amrik Singh Nimbran

B3-304, Palm Grove Heights

Ardee City

Sector 52, Gurugram

Haryana

India 122003

Paul Levrie

Faculty of Applied Engineering

University of Antwerp

Groenenborgerlaan 171

2020 Antwerpen

and

Department of Computer Science

KU Leuven

P. O. Box 2402

3001 Heverlee

Belgium

**Abstract:**

We examine the structure of the periodic continued fractions of square
roots of non-square positive integers given by an integer-valued quadratic
polynomial
*Q*(*n*) = (*an* + *b*)^{2} + (*gn* + *h*). The aim is to identify repeated
blocks of partial quotients in the period. The quotients in the period
form a palindrome, and when the period length is even, the period has
a central term *a*_{n}.
The paper focuses on periods with
*a*_{n} = *a*_{0} or
*a*_{n} = *a*_{0} – 1,
where *a*_{0} is the initial partial quotient.
For *a*_{n} = *a*_{0} we give an algorithm
to obtain formulas involving repeated blocks comprising three or more
elements, not all equal.

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(Concerned with sequences
A000129
A001333
A002530
A002531
A014106
A041006
A041007
A041010
A041011
A041014
A041015
A041030
A041031
A041066
A041067
A056220
A080856
A081585
A142463.)

Received August 17 2022; revised versions received September 21 2022; October 12 2022.
Published in *Journal of Integer Sequences*,
March 2 2023.

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