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The Equation
(***j*+*k*+1)^{2}-4*k* = *Qn*^{2}
and Related Dispersions

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Clark Kimberling

Department of Mathematics

University of Evansville

1800 Lincoln Avenue

Evansville, IN 47722

USA

**Abstract:**

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}-4*k* = *Qn*^{2} .
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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