[seqfan] Re: Number of points with maximum norm n in A2, A3, A4, A5, D5
Richard Mathar
mathar at strw.leidenuniv.nl
Fri Feb 5 00:41:53 CET 2010
Putting the numbers in
http://list.seqfan.eu/pipermail/seqfan/2010-February/003582.html
into concrete with some explicit summation over the polytopes
shows that the sequence of the number of lattice points
in the A5 lattice with maximum norm n is given by
a(0)=1
a(n>0) = 50*n^2+88*n^4+2
= 1, 140, 1610, 7580, 23330, 56252, 115850, 213740, 363650, 581420, 885002,...
The equivalent sequence for the D5 lattice becomes
a(0)=1
a(n>0) = 80n^4+40*n^2+1+(-1)^n
= 1, 120, 1442, 6840, 21122, 51000, 105122, 194040, 330242, 528120, 804002, 1176120, 1664642...
Both are in agreement with the brute force enumeration within the slabs.
Recurrences and generating functions follow as usual.
Richard
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