Alcuin's Sequence

The Integer Sequence 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, 19, ... (Sloane's A005044) given by the Coefficients of the Maclaurin Series for $1/(1-x^2)(1-x^3)(1-x^4)$. The number of different Triangles which have Integral sides and Perimeter $n$ is given by

$$T(n) = P_3(n)-\sum_{1\leq j\leq\left\lfloor{n/2}\right\rfloor } P_2(j)$$ (1)

$$= \left[{n^2\over 12}\right]-\left\lfloor{n\over 4}\right\rfloor \left\lfloor{n+2\over 4}\right\rfloor$$ (2)

$$= \left\{\begin{array}{ll} \left[{n^2\over 48}\right]& \mbox{for $n$\ even}\\ \left[{(n+3)^2\over 48}\right]& \mbox{for $n$\ odd,}\end{array}\right.$$ (3)

where $P_2(n)$ and $P_3(n)$ are Partition Functions, with $P_k(n)$ giving the number of ways of writing $n$ as a sum of $k$ terms, $[x]$ is the Nint function, and $\left\lfloor{x}\right\rfloor$ is the Floor Function (Jordan et al. 1979, Andrews 1979, Honsberger 1985). Strangely enough, $T(n)$ for $n=3$, 4, ... is precisely Alcuin's sequence.

See also Partition Function P, Triangle

References

Andrews, G. ``A Note on Partitions and Triangles with Integer Sides.'' Amer. Math. Monthly 86, 477, 1979.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 39-47, 1985.

Jordan, J. H.; Walch, R.; and Wisner, R. J. ``Triangles with Integer Sides.'' Amer. Math. Monthly 86, 686-689, 1979.

Sloane, N. J. A. Sequence A005044/M0146 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.