Closed form expressions for the iterated floor function

For positive integers k, r, and n ⩾ k + 1, the iterated floor function fk,r is defined by fk,r(k + 1) = r;fk,r(n)= ⌊nn−kfk,r(n−1) ⌋, n>k+1.A special case (k = r = 3) of this function occurs as an upper bound on the number of 3-subsets, excluding tetrahedra, of an n-set (Turàn's problem). For certain values of k and r, this note establishes closed form expressions for fk,r, then uses them to prove some interesting properties.