 TOPICS  # 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, ... (OEIS A005044) given by the coefficients of the Maclaurin series for (1) A binary plot of the first few terms in the sequence is illustrated above.

Closed forms include   (2)   (3)   (4)

where is the floor function.

The number of different triangles which have integral sides and perimeter is given by   (5)   (6)   (7)

where and are partition functions, with giving the number of ways of writing as a sum of terms, is the nearest integer function, and is the floor function (Jordan et al. 1979, Andrews 1979, Honsberger 1985). Strangely enough, for , 4, ... is precisely Alcuin's sequence.

Integer Triangle, Partition Function P, Triangle Dissection

## Explore with Wolfram|Alpha ## 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 "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Alcuin's Sequence

## Cite this as:

Weisstein, Eric W. "Alcuin's Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AlcuinsSequence.html