Notes on Symmetric Bases

Abstract

A_k = {1, a_2, ... a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i. An extremal h-basis A_k is one for which n is as large as possible. Computing extremal bases is known as the Postage Stamp Problem. A basis A_k is symmetric if A_k = {1, a_2, ... a_k} where a_i + a_(k-i) = a_k for 1<=i<=k-1. Examination of a number of symmetric bases suggests the following conjecture: if the range 0 ... a_k is covered using at most h stamps, then the range 0 ... ha_k is also covered using at most h stamps. This paper shows that this is not strictly true, but demonstrates that there is a value h_1 such that the conjecture is true for all h>=h_1.