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In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of:[1]

x3(1x2)(1x3)(1x4)=x3+x5+x6+2x7+x8+3x9+.

The sequence begins with these integers:[1][2]

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.[2] It is also the number of triangles with distinct integer sides and perimeter n + 6, i.e. number of triples (abc) 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.

References

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  2. 2.0 2.1 Template:SloanesRef