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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:

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

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.

References

Weisstein, Eric W., "Alcuin's Sequence", MathWorld.
"Sloane's A005044 : Alcuin's sequence", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Mathematics Encyclopedia