上限函数和解嵌套递归
Sums of Ceiling Functions Solve Nested Recursions
摘要：
已知对于给定的整数s \geq 0和j > 0，嵌套递归R（n）= R（n-s- R（n-j））+ R（n -2 j-s-R（n -3 j））具有封闭形式的解，对于该解存在无限标号树的组合解释.当s = 0时，我们证明了这个解序列有一个封闭的形式，它是天花板函数C（n）的和.进一步，在适当的初始条件下，我们得到了关于参数s1，a1，s2和a2的充要条件，使得C（n）解嵌套递归R（n）= R（n-s1- R（n-a1））+ R（n-s2- R（n-a2））.
英文摘要：
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)).
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