Partial complements and transposable dispersions

Kimberling, Clark ; Brown, John E.

Journal of integer sequences, Tome 7 (2004) no. 1

Résumé

Suppose $A=\{a(i,j)\},$ for $i\geq 0$ and $j\geq 0,$ is the dispersion of a strictly increasing sequence $r=(r(0),r(1),r(2),\ldots )$ 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 $n\geq 0.$ 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 $i+j\theta $. For fixed $i\geq 0,$ the sequence $a(i,j)$, for $j\geq 1,$ 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 $t^{(2$m$)}r$ converge to a sequence in $F$.

Détail

Classification : 11B36, 11B25, 11B37
Keywords: central polygonal numbers, dispersion, interspersion, Farey sequence, Farey tree, fractal sequence, partial complement, polygonal numbers, rank array, rank sequence