Nested Recursions with Ceiling Function Solutions

Unless otherwise noted, we consider only n > 0. The parameters in (1.1) are all integers satisfying k, pi and aij > 0. Assume c initial conditions R(1) = ξ1, R(2) = ξ2, . . . , R(c) = ξc, with all ξi > 0. Golomb [6] first solved the simplest example of such a non-homogeneous nested recursion, namely, G(n) = G(n− G(n− 1)) + 1, G(1) = 1; see also [7]. In fact, all of the recursions we find with ceiling function solutions can be rewritten into an equivalent homogeneous form. We use the non-homogeneous form for two reasons; first, we do not a priori know we will find no others, and second, it simplifies the presentation of the material in Section 2.