Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.7

The Equation (j+k+1)2-4k = Qn2 and Related Dispersions

Clark Kimberling
Department of Mathematics
University of Evansville
1800 Lincoln Avenue
Evansville, IN 47722


Suppose Q is a positive nonsquare integer congruent to 0 or 1 mod 4. Then for every positive integer n, there exists a unique pair (j,k)$ of positive integers such that (j+k+1)2-4k = Qn2 . This representation is used to generate the fixed-j array for Q and the fixed-k array for Q. These arrays are proved to be dispersions; i.e., each array contains every positive integer exactly once and has certain compositional and row-interspersion properties.

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(Concerned with sequences A005752 A033313 A033317 A038150 A077428 A073855 A080164 A087076 A087079 A098021 A120858 A120859 A120860 A120861 A120862 A120863 A120864 A120865 A120866 A120867 A120868 A120869 A120870 A120871 A120872 A120873 and A120874 .)

Received July 10 2006; revised version received February 20 2007. Published in Journal of Integer Sequences, March 13 2007.

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