Alcuin's Sequence

Alcuin's sequence counts triangles with positive integer side lengths by perimeter. For perimeters 0, 1, 2, ..., it begins 0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, ... (OEIS A005044). Writing for the count at perimeter , the shifted counts are 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, .... These are the coefficients of the Maclaurin series for

(1)

A binary plot of the first few terms in the sequence is illustrated above.

Closed forms include

			(2)
			(3)

where is the floor function.

The number of different triangles which have integral sides and perimeter is given by

			(4)
			(5)
			(6)

where and are partition functions, with giving the number of ways of writing as a sum of positive integers, is the nearest integer function, and is the floor function (Andrews 1979, Jordan et al. 1979, Honsberger 1985). The count for , 4, ... is precisely the shifted list displayed above, with .