The maximal solution of a restricted subadditive inequality in numerical analysis

For a fixed non-negative integerp, letU _2p = {U _2p (n)},n ≥ 0, denote the sequence that is defined by the initial conditionsU _2p (0) =U _2p (1) =U _2p (2) = =U _2p (2p) = 1 and the restricted subadditive recursion $$U_{2p} (n + 2p + 1) = \mathop {\min }\limits_{0 \leqslant l \leqslant p} (U_{2p} (n + l) + U_{2p} (n + 2p - l)),n \geqslant 0$$ U _2p is of importance in the theory of sequential search for simple real zeros of real valued continuous 2p-th derivatives In this paper, several closed form expressions forU _2p (n), n > 2p, are determined, thereby providing insight into the structure ofU _2p Two of the properties thus illuminated are (a) the existence of exactlyp + 1 limit points (1 + 1/(p + 1 +i), 0 ≤i ≤p) of the associated sequence {U _2p (n + 1)/U _2p (n)},n ≥ 0, and (b) the relevance toU _2p of the classic number theoretic function ord