The maximal solution of a restricted subadditive inequality in numerical analysis

Abstract

For a fixed non-negative integerp, letU2p = {U2p(n)},n ≥ 0, denote the sequence that is defined by the initial conditionsU2p(0) =U2p(1) =U2p(2) = =U2p(2p) = 1 and the restricted subadditive recursion $$U_{2p} (n + 2p + 1) = \mathop {\min }\limits_{0 \leqslant l \leqslant p} (U_{2p} (n + l) + U_{2p} (n + 2p - l)),n \geqslant 0$$ U2p 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 forU2p(n), n > 2p, are determined, thereby providing insight into the structure ofU2p 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 {U2p(n + 1)/U2p(n)},n ≥ 0, and (b) the relevance toU2p of the classic number theoretic function ord