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Search: id:A140439
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| A140439 |
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Array, by antidiagonals, arises in counting <= k facets in d-dimensional n-point sets. |
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1
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| 3, 4, 9, 5, 16, 18, 6, 25, 40, 30, 7, 36, 75, 80, 45, 8, 49, 126, 175, 140, 63, 9, 64, 196, 336, 350, 224, 84, 10, 81, 288, 588, 756, 630, 336, 108, 11, 100, 405, 960, 1470, 1512, 1050, 480, 135
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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The formula in the cited paper, p. 10, is actually given as (d+1)*C((k+d),d) with the restriction k < floor(n/(d+1)), but that restriction is obviated in the conclusion as an artefact of the proof. The formula is a lower bound for counting <= k facets in d-dimensional n-point sets in R^d. In the table shown, the first column is (d+1) and the second column is (d+1)^2.
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LINKS
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Oswin Aichholzer, Jesus Garcia, David Orden and Pedro Ramos, New results on lower bounds for the number of (at most k)-facets
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FORMULA
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A[d,n] = (d+1)*C((k+n),n).
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EXAMPLE
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Table begins:
===============================================================================
...|.n=0.|.n=1.|.n=2.|..n=3.|..n=4.|...n=5.|...n=6.|...n=7.|....n=8.|....n=9.|.in.OEIS
===============================================================================
k=2.|...3.|...9.|..18.|...30.|...45.|....63.|....84.|...108.|....135.|....165.|A045943
k=3.|...4.|..16.|..40.|...80.|..140.|...224.|...336.|...480.|....660.|....880.|4*A000292
k=4.|...5.|..25.|..75.|..175.|..350.|...630.|..1050.|..1650.|...2475.|...3575.|
k=5.|...6.|..36.|.126.|..336.|..756.|..1512.|..2772.|..4752.|...7722.|..12012.|
k=6.|...7.|..49.|.196.|..588.|.1470.|..3234.|..6468.|.12012.|..21021.|..35035.|
k=7.|...8.|..64.|.288.|..960.|.2640.|..6336.|.13728.|.27456.|..51480.|..91520.|
k=8.|...9.|..81.|.405.|.1485.|.4455.|.11583.|.27027.|.57915.|.115830.|.218790.|
===============================================================================
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CROSSREFS
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Cf. A000292, A045943.
Sequence in context: A011292 A021745 A070154 this_sequence A023183 A102320 A021290
Adjacent sequences: A140436 A140437 A140438 this_sequence A140440 A140441 A140442
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 20 2008
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