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Alcuin's Sequence

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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/((1-x^2)(1-x^3)(1-x^4))=1+x^2+x^3+2x^4+x^5+....
(1)
AlcuinsSequence

A binary plot of the first few terms in the sequence is illustrated above.

Closed forms include

a_n=1/(288){107+54n+6n^2+(-1)^n[81+18n+64cos(1/3pin)]+36[cos(1/2pin)-sin(1/2pin)]}
(2)
=1/(288){107+54n+6n^2+(-1)^n[81+18n+64cos(1/3pin)]
(3)
+36(-1)^(|_(n+1)/2_|),
(4)

where |_x_| is the floor function.

The number of different triangles which have integral sides and perimeter n is given by

T(n)=P_3(n)-sum_(1<=j<=|_n/2_|)P_2(j)
(5)
=[(n^2)/(12)]-|_n/4_||_(n+2)/4_|
(6)
={[(n^2)/(48)] for n even; [((n+3)^2)/(48)] for n odd,
(7)

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 nearest integer function, and |_x_| 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

Integer Triangle, Partition Function P, Triangle Dissection

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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

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