Question 19
A sequence $a_0,a_1,a_2,a_3,\ldots$, with $a_1=1$, is defined such that for any positive integers $m,n$, where $m\ge n$, the terms of the sequence satisfy the relation $$a_{m+n}+a_{m-n}+m-n=\frac12(a_{2m}+a_{2n})+1.$$ Find the value of $\left\lfloor\frac{a_{2014}}{2014}\right\rfloor$.