Sums of Ceiling Functions Solve Nested Recursions
نویسندگان
چکیده
It is known that, for given integers s ≥ 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) = ∑j−1 i=0 ⌈
منابع مشابه
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 c...
متن کاملNested Recursions, Simultaneous Parameters and Tree Superpositions
We apply a tree-based methodology to solve new, very broadly defined families of nested recursions of the general form R(n) = ∑k i=1R(n − ai − ∑p j=1R(n − bij)), where ai are integers, bij are natural numbers, and k, p are natural numbers that we use to denote “arity” and “order,” respectively, and with some specified initial conditions. The key idea of the tree-based solution method is to asso...
متن کاملFinding efficient recursions for risk aggregation by computer algebra
We derive recursions for the probability distribution of random sums by computer algebra. Unlike the well-known Panjer-type recursions, they are of finite order and thus allow for computation in linear time. This efficiency is bought by the assumption that the probability generating function of the claim size be algebraic. The probability generating function of the claim number is supposed to b...
متن کاملUMMER – Transcendental Functions and Symbolic Summation in F ORM
Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums, where the harmonic sums and their generalizations appear as building blocks, originating for example from the expansion of generalized hypergeometric functions around integer values of the par...
متن کاملA new method for fuzzification of nested dummy variables by fuzzy clustering membership functions and its application in financial economy
In this study, the aim is to propose a new method for fuzzification of nested dummy variables. The fuzzification idea of dummy variables has been acquired from non-linear part of regime switching models in econometrics. In these models, the concept of transfer functions is like the notion of fuzzy membership functions, but no principle or linguistic sentence have been used for inputs. Consequen...
متن کامل