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Search: id:A124278
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| A124278 |
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Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n. |
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2
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| 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 3, 4, 4, 3, 2, 1, 1, 2, 5, 5, 4, 3, 2, 1, 1, 4, 7, 8, 6, 5, 3, 2, 1, 1, 3, 8, 9, 9, 6, 5, 3, 2, 1, 1, 5, 11, 14, 12, 10, 7, 5, 3, 2, 1, 1, 4, 12, 16, 16, 13, 10, 7, 5, 3, 2, 1, 1, 7, 16, 23, 22, 19, 14, 11, 7, 5, 3, 2, 1, 1, 5, 18, 25, 28, 24
(list; table; graph; listen)
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OFFSET
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3,11
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COMMENT
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For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides. Column k=3 is A005044, column k=4 is A062890, column k=5 is A069906 and column k=6 is A069907.
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LINKS
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T. D. Noe, Rows n=3..102 of triangle, flattened
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-Gon Partitions, Bull. Austral. Math. Soc. 64 (2001), 321-329.
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FORMULA
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G.f. for column k is x^k/(product_{i=1..k} 1-x^i) - x^(2k-2)/(1-x)/(product_{i=1..k-1} 1-x^(2i))
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EXAMPLE
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For polygons having perimeter 7: 2 triangles, 2 quadrilaterals, 2 pentagons, 1 hexagon and 1 heptagon. The triangle begins
1
0 1
1 1 1
1 1 1 1
2 2 2 1 1
1 3 2 2 1 1
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MATHEMATICA
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Needs["DiscreteMath`Combinatorica`"]; Table[p=Partitions[n]; Length[Select[p, Length[ # ]==k && #[[1]]<Total[Rest[ # ]]&]], {n, 3, 30}, {k, 3, n}]
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CROSSREFS
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Cf. A124287 (similar, but with no restriction on the sides).
Sequence in context: A135265 A144110 A076490 this_sequence A139755 A104637 A058745
Adjacent sequences: A124275 A124276 A124277 this_sequence A124279 A124280 A124281
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KEYWORD
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nice,nonn,tabl
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Oct 24 2006
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