I am interested in the efficient computability of sequences.

Is it possible some ``interesting sequences'' be computed via addition formulae/semigroup operation?

Here is an artificial example.

Suppose one finds an associative operation $f : \mathbb{Z}^2 \times \mathbb{Z}^2 \to \mathbb{Z}^2$ and a sequence $a_n$ such that:

$f([a_{2n},a_{2n+1}],[a_{2m},a_{2m+1}]) = [ a_{2n+2m},a_{2n+2m+1} ]$

$f$ is associative and the result is in [$a_{2k},a_{2k+1}$] which resembles a semigroup

Let $a_n = A000069(n)$ where A000069 Odious numbers: numbers with an odd number of 1's in their binary expansion.

From the OEIS comment: $a_{2n+1} + a_{2n} = A017101(n) = 8n+3$

one can find $n,m$.

From: $a_n = \frac{1}{2} (4n + 1 + (-1)^{A000120(n)})$

A000120 1's-counting sequence: number of 1's in binary expansion of n

one can find

[ $a_{2n+2m},a_{2n+2m+1}$ ]

and this seems to complete associativity and closure.

So is [A000069(2n),A000069(2n+1)] $ \in \mathbb{Z}^2 $a semigroup?

If yes what type of semigroup is it (the semigroup operation involves counting ones in the binary expansion)?

Are there other similar examples/constructions of sequences that are not rational functions or related to multiples of points on curves (e.g. Fibonacci, Somos4).