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Alcuin,s Sequence

المؤلف:  Andrews, G

المصدر:  "A Note on Partitions and Triangles with Integer Sides." Amer. Math. Monthly 86

الجزء والصفحة:  ...

29-12-2020

1325

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

		{107+54n+6n^2+(-1)^n[81+18n+64cos(1/3pin)]+36[cos(1/2pin)-sin(1/2pin)]}" src="https://mathworld.wolfram.com/images/equations/AlcuinsSequence/Inline3.gif" style="height:23px; width:506px" />	(2)
		{107+54n+6n^2+(-1)^n[81+18n+64cos(1/3pin)]" src="https://mathworld.wolfram.com/images/equations/AlcuinsSequence/Inline6.gif" style="height:23px; width:329px" />	(3)
			(4)

where  is the floor function.

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

			(5)
			(6)
		{[(n^2)/(48)] for n even; [((n+3)^2)/(48)] for n odd," src="https://mathworld.wolfram.com/images/equations/AlcuinsSequence/Inline20.gif" style="height:92px; width:133px" />	(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.

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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."