For r=sqrt(n), the solutions to x^2=ceiling(x*r*floor(x/r)) appear to be given by the Chebyshev polynomial T(k,x) k>0 evaluated at x=a(n). These solutions also seem to be given by a sequence (b(k))_(k>=0) satisfying the recurrence b(k)= 2*A002350(n)*b(k-1)-b(k-2).