Closed form expressions for the iterated floor function

Abstract

For positive integers k, r, and n ⩾ k + 1, the iterated floor function f_k,r is defined by

$$\text{f}{}_{\mathrm{k,r}}\text{(k + 1) = r;}$$

f_k,r(n)=

$$\text{⌊}\text{n}\text{n−k}\text{f}{}_{\mathrm{k,r}}\text{(n−1) ⌋, n>k+1}$$

.

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 f_k,r, then uses them to prove some interesting properties.