Problem:
In the non-decreasing sequence of odd integers {a1​,a2​,a3​,…}={1,3,3,3,5,5,5,5,5,…} each positive odd integer k appears k times. It is a fact that there are integers b,c and d such that, for all positive integers n,
an​=b[n+c​]+d,
where [x] denotes the largest integer not exceeding x. The sum b+c+d equals
Therefore, for some integers m and k,2+c=k2 and 5+c=m2. Since the only perfect squares that differ by 3 are 1 and 4,k=1 and m=2. Therefore c=−1 and d=1. These are necessary (and as it turns out also, sufficient) conditions.