[seqfan] Addition formulae for sequences and semigroups

[Georgi Guninski]

Suppose one finds a function f 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, how different from a semigroup is it?

Take for example A000069  Odious numbers: numbers with an odd number of 1's in their binary expansion. 

>From the OEIS comment:
a(2*n+1) + a(2*n) = A017101(n) = 8*n+3

one can find n,m.

From:
a(n) = 1/2 * (4n + 1 + (-1)^A000120(n))

one can find [ a(2n+2m),a(2n+2m+1) ]

well, i admit finding n,m is borderline cheating but couldn't find better example excluding rational expressions and curves.