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
Thus, subtracting the first equation from the second
2=b([2+c]−[1+c]).
Since the quantity in parentheses is at most one, b=2. Then by subtractions,
1+[1+c]=[2+c]=[3+c]=[4+c]=[5+c]−1.
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.