Journal of Integer Sequences, Vol. 26 (2023), Article 23.2.8

Patterns in Continued Fractions of Square Roots

Amrik Singh Nimbran
B3-304, Palm Grove Heights
Ardee City
Sector 52, Gurugram
India 122003

Paul Levrie
Faculty of Applied Engineering
University of Antwerp
Groenenborgerlaan 171
2020 Antwerpen
Department of Computer Science
KU Leuven
P. O. Box 2402
3001 Heverlee


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 an. The paper focuses on periods with an = a0 or an = a0 – 1, where a0 is the initial partial quotient. For an = a0 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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