Question

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.

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

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

3. 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).

Answer 1

Is your functional equation a non-vacuous constraint? Since summation is group operation, the pairs [2n,2n+1] form a group 2N, which map to your group [a(2n),a(2n+1)]. For example, for A000069

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, ...
0  1  2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  
   ^  ^     ^           ^   ^           ^       ^   ^       ^           ^       ^   
----  ----  -----   -----   -----  ------  ------   -----   -----  ------  ------

each pair [2n,2n+1] is a group element is, so that we have

[a(6),a(7)] + [a(10),a(11)] = [a(16),a(17)]

and yet there seems to be no relation among a(6), a(10) and a(16).