My wife has bought a flock of hens to have some eggs for this year’s Easter and I have to feed them. On day 0, they got 10 kgs of wheat, the next day (day 1), I gave them 14 kgs. Since then, I give them the average of the previous two days increased by 1 kg in even numbered days and decreased by 1 kg in odd numbered days.
I have $$a_n=\frac{a_{n-1} + a_{n-2}}2 +(-1)^n$$ but I don't know how to proceed or if it's correct.
The sequence converges to $o(a_0,a_1)$ for the odd numbers and to $o(a_0,a_1)+2$ for the even numbers, where $$o(a_0,a_1)=\frac{a_0+2a_1-2}3.\tag 1$$
To be honest, I don't know why.
EDIT
The only reason behind the assertion above is that if the odd day numbers and the even day numbers converge then the following equation has to hold
$$o=\frac{a_0+2a_1-2}3=\frac{o+e}2-1=\frac{\frac{a_o+2a_1-2}3+\frac{a_o+2a_1-2}3+2}2-1$$ and it does.
EDIT 2
But to say "it does" is not enough. Because the following identities hold as well
$$o=\frac{\frac{\alpha a_o+\beta a_1-2}3+\frac{\alpha a_o+\beta a_1-2}3+2}2-1$$ for $o=\frac{\alpha a_0+\beta a_1-2}3.$
Yet the limit IS $(1)$!