Alcuin's sequence

In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of:

${\frac {x^{3}}{(1-x^{2})(1-x^{3})(1-x^{4})}}=x^{3}+x^{5}+x^{6}+2x^{7}+x^{8}+3x^{9}+\cdots .$

The sequence begins with these integers:

0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21

The nth term is the number of triangles with integer sides and perimeter n. It is also the number of triangles with distinct integer sides and perimeter n + 6, i.e. number of triples (a, b, c) such that 1 ≤ a < b < c < a + b, a + b + c = n + 6.

If one deletes the three leading zeros, then it is the number of ways in which n empty casks, n casks half-full of wine and n full casks can be distributed to three persons in such a way that each one gets the same number of casks and the same amount of wine.

The sequence with the three leading zeros deleted is obtained as the sequence of coefficients of the power-series expansion of

${\frac {1}{(1-x^{2})(1-x^{3})(1-x^{4})}}=1+x^{2}+x^{3}+2x^{4}+x^{5}+3x^{6}+\cdots .$

This sequence has also been called Alcuin's sequence by some authors.