OFFSET
1,1
COMMENTS
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 artifact 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.
LINKS
Oswin Aichholzer, Jesús García, David Orden and Pedro Ramos, New results on lower bounds for the number of (at most k)-facets arXiv:0801.1036
FORMULA
A[d,n] = (d+1)*C((k+n),n).
EXAMPLE
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.|
===============================================================================
CROSSREFS
KEYWORD
AUTHOR
Jonathan Vos Post, Jun 20 2008
STATUS
approved