Suppose that $ a_0 < a_1,$ $ b_0 < b_1,$ and $ $ a_n=a_1b_{n1}+a_0b_{n2}+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_na_{n1},b_nb_{n1})\},$ $ 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_na_{n1})$ ever linearly recurrent?

Let $ d$ be a number that occurs infinitely many times in $ (b_nb_{n1})$ , and let $ (p_n)$ be the sequence of numbers $ k$ such that $ b_kb_{k1}=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}.$