Sums of Ceiling Functions Solve Nested Recursions
Indexed on: 19 Jul '11Published on: 19 Jul '11Published in: Mathematics - Combinatorics
Abstract
It is known that, for given integers s \geq 0 and j > 0, the nested recursion R(n) = R(n - s - R(n - j)) + R(n - 2j - s - R(n - 3j)) has a closed form solution for which a combinatorial interpretation exists in terms of an infinite, labeled tree. For s = 0, we show that this solution sequence has a closed form as the sum of ceiling functions C(n). Further, given appropriate initial conditions, we derive necessary and sufficient conditions on the parameters s1, a1, s2 and a2 so that C(n) solves the nested recursion R(n) = R(n - s1 - R(n - a1)) + R(n- s2 - R(n - a2)).
