# The algebraic properties of a sequence

Take the sequence $$S$$ to be $$4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13,...$$. Clearly the odd indices of the sequence are elements of $$4\mathbb{N}^+$$, so the odd indices of $$S$$ form a group without inverses or the identity element. The even indices of the sequence are the odd numbers greater than 1.

We have $$S_a + S_b -1 = S_{a+b}$$ where $$a, b$$ are even indices of $$S$$. I am trying to find a more general set of operations/identities over $$S$$ such that $$S_x \star S_y = S_{f(x, y)}$$ for any indices $$x, y$$ for example. Is this possible? How would I go about defining something like this?

I am learning about groups/rings/fields right now and trying to apply what I am learning to this, so I am wondering how this problem connects to this area, if it even does.

• @Shaun $a, b$ are even indices. – Grassi Apr 21 at 0:16
• This is not clear. Since, say, the even numbered terms are just the odd numbers starting from $3$, so long as you stay within the odd numbers you can define $f(a,b)$ by the relation you specify. Similar comment for the odd numbered terms so long as you stay in multiples of $4$. What are you hoping for? – lulu Apr 21 at 0:36
• And, just to say, referring to "a group without inverses or identity" is like referring to a bicycle with no wheels or frame. I think all you mean here is a subset of the natural numbers which is closed under addition. – lulu Apr 21 at 0:40
• "a group without inverses or the identity" is called a "semigroup"... – Arturo Magidin Apr 21 at 0:44