en>Sfan00 IMG at 14:35, 20 April 2013

2013-04-20T14:35:27Z

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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:<ref name=wolfram>{{MathWorld|title = Alcuin's Sequence |id = AlcuinsSequence}}</ref>

: <math> \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. </math>

The sequence begins with these integers:<ref name=wolfram/><ref name=oeis>{{SloanesRef |sequencenumber=A005044 |name=Alcuin's sequence}}</ref>

: 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 ''n''th term is the number of [[Heron triangle|triangles with integer sides]] and perimeter ''n''.<ref name=oeis/> 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==

{{reflist}}

[[Category:Integer sequences]]

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en>Sfan00 IMG at 14:35, 20 April 2013