Finite sums of the Alcuin numbers

摘要

The Alcuin number t(n) is equal to the number of incongruent integer triangles having perimeter n, where n is an integer. Using generating functions, we give a derivation of well-known formulas for the Alcuin sequence {t(n)} that involves the closest integer function {double pipe}x{double pipe}, and floor function ?x?. These formulas do not lend themselves very easily to operations such as summation. To find the number of incongruent integer triangles having perimeter at most n, we must evaluate the sum ∑~n_(k=0) t(k), or to find those with even perimeter up to 2n, we must evaluate ∑~n_(k=0) t(2k). These computations are theoretically awkward. In this article, we develop formulas in both closed form and abbreviated form for these sums using generating functions. In the process, we exploit the relationship between the Alcuin numbers and partitions of integers.