I'm trying to prove that the semigroup $<F_{i}, F_{i+3}, F_{i+3n}>$ is equivalent to the semigroup $<F_{i}, F_{i+3}>$. From my understanding, a numerical semigroup is a specific kind of semigroup, where the binary operation in this case is addition.
I'm quite sure that the result is true that the two semigroups are equivalent, but I'm not quite sure how to go about proving it. I've tried to carry out the same method that has been used to show that these two semigroups: $<F_{i}, F_{i+1}, F_{i+k}>$ and $<F_{i}, F_{i+1}>$ are equal through addition of Fibonaaci numbers, but it does not seem to work due to $F_{i+3}$ being three terms apart from $F_{i}$. Can anyone help me with a proof for this?
We have $$F_{i+3}=F_{i+2}+F_{i+1}=2F_{i+1}+F_i\,,\\ F_{i+4}=F_{i+3}+F_{i+2}=F_{i+3}+F_{i+1}+F_i\,,\\ F_{i+5}=F_{i+4}+F_{i+3}=2F_{i+3}+F_{i+1}+F_i\,,\\ F_{i+6}=F_{i+5}+F_{i+4}=3F_{i+3}+2F_{i+1}+2F_i\,. $$ So, using the first line, $F_{i+6}=4F_{i+3}+F_i$ which is in the subsemigroup generated by $F_i$ and $F_{i+3}$.
Then use induction.