Suppose that $a_0 < a_1,$ $b_0 < b_1,$ and $$a_n=a_1b_{n-1}+a_0b_{n-2}+qn+r$$ for $n \geq 2$, where $a_0,a_1,b_0,b_1,q,r$ are integers such that $(a_n)$ and $(b_n)$ are increasing and ${(|a_n|)}$ and ${(|b_n|)}$ partition the positive integers. What can be proved about the cardinality of $$D=\{(a_n-a_{n-1},b_n-b_{n-1})\},$$ for $n \geq 0?$

Experimental results:

If $(a_0,a_1,b_0,b_1,q,r)=(-1,2,3,4,2,0)$, then $|D|=9$; see "Experimental fact" at A possibly surprising appearance of $\sqrt{2}.$

If $(a_0,a_1,b_0,b_1,q,r)=(1,2,3,4,1,0)$, then $D=\{(1,1),(4,1),(4,2),(5,1),(6,1),(11,1)\}.$

If $(a_0,a_1,b_0,b_1,q,r)=(3,4,1,2,1,-7)$, then $D=\{(1,1),(2,3),(8,1),(8,2),(11,1),(12,1),(16,2),(18,1)\}.$

Reasons for studying the set $D$ include these related questions:

Is $(a_n-a_{n-1})$ ever linearly recurrent?

Let $d$ be a number that occurs infinitely many times in $(b_n-b_{n-1})$, and let $(p_n)$ be the sequence of numbers $k$ such that $b_k-b_{k-1}=d.$ Must $(p_n/n)$ converge? As an example, for $(a_0,a_1,b_0,b_1,q,r)=(-1,2,3,4,2,0)$, we have $$(p_n) = (1,11,13,16,19,22,25,28,31,34,37,43,45,51,53,56,62,\dots),$$ and it appears that $\lim_{n \to \infty} p_n/n = 1+\sqrt{2}.$