Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.6 |

Department of Mathematics

University of Evansville

1800 Lincoln Avenue

Evansville, IN 47722

USA

**Abstract:**

Suppose
for
and
is the dispersion of a
strictly increasing sequence
of integers, where *r*(0)=1 and infinitely many postive integers are not terms of *r*. Let *R* be the set of such sequences, and define *t* on *R* by
*tr*(*n*)=*a*(0,*n*) for
Let *F* be the subset of *R* consisting of sequences *r* satisfying *ttr*=*r*. The set *F* is characterized in terms of ordered
arrangements of numbers .
For fixed
the sequence
*a*(*i*,*j*), for
is the (*i*+1)st partial complement of *r*. Central to the characterization of *F* is the role of the families of
figurate (or polygonal) number sequences and the centered polygonal number
sequences. Finally, it is conjectured that for every *r* in *R*, the
iterates *t*^{(2m)}*r* converge to a sequence in *F*.

(Concerned with sequences A000124 A000217 A000290 A000326 A001401 A001840 A001844 A001972 A002061 A002620 A005448 A008748 A077043 A082152 A082153 A082154 A082155 A082156 A086270 A086271 A086272 A086273 A087465 A087466 A087467 A087468 A087469 A087470 .)

Received October 14 2003;
revised version received January 19 2004.
Published in *Journal of Integer Sequences* February 19 2004.

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