A study of a family of self-referential sequences

Abstract: We introduce and analyze a three-parameter family of self-referential integer sequences $S(x,y,z)$: starting from $a(1)=x$, each term advances by $y$ when the index $k$ has already appeared as a value and by $z$ otherwise. This simple rule generates a surprising zoo of behaviors, many of which are catalogued -- albeit in a rather unstructured fashion -- in the OEIS. Whenever $y>z>0$, we prove that the density $a(k)/k$ converges to the positive root of $r^2 - z r - (y - z) = 0$. Two subfamilies, $S(x,Z+1,Z)$ and $S(x,Z,Z+1)$, yield explicit non-homogeneous Beatty sequences, providing explicit formulas for numerous OEIS entries. For $y=0$ and $z \ge 2$, the sequences eventually become periodic and satisfy linear recurrences. Critical cases with a zero discriminant unveil geometric patterns on triangular, square, and hexagonal lattices. Finally, via tree-like representations we uncover a tight link with meta-Fibonacci recurrences.