OFFSET
0,2
COMMENTS
From Robert Israel, May 03 2019: (Start)
a(n) is the number of integer lattice points in n*C where C is the polytope in R^(4 X 4) defined by Sum_{1<=i<=4} x_{i,j} = Sum_{1<=i<=4} x_{j,i} = Sum_{1<=i<=4} x_{i,1} for 1<=j<=4 and 0 <= x_{i,j} <= 1 for 1<=i,j<=4.
The vertices of this polytope have coordinates in {0,1/2,1} (an example of a vertex with non-integer coordinates is [0,1,1,1/2; 1,0,1,1/2; 1,1,0,1/2; 1/2,1/2,1/2,1]).
Therefore a(n) should be quasi-polynomial in n. (End)
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..40 (terms 1..21 from R. H. Hardin)
Wikipedia, Ehrhart quasi-polynomials
FORMULA
Conjecture: a(n) = (29 + 3*(-1)^n)/32 + (34/7)*n + (7202/525)*n^2 + (4658/189)*n^3 + (118873/3780)*n^4 + (5321/180)*n^5 + (36827/1800)*n^6 + (1285/126)*n^7 + (17581/5040)*n^8 + (2789/3780)*n^9 + (2789/37800)*n^10. - Robert Israel, May 03 2019
EXAMPLE
Some solutions for n=3
..2..2..1..3....2..1..2..1....3..2..1..0....1..3..2..2....0..3..2..2
..1..2..3..2....1..3..2..0....0..1..2..3....2..2..2..2....1..2..3..1
..3..1..3..1....2..1..1..2....1..1..1..3....2..1..2..3....3..2..1..1
..2..3..1..2....1..1..1..3....2..2..2..0....3..2..2..1....3..0..1..3
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Dec 24 2011
EXTENSIONS
a(0)=1 prepended by Andrew Howroyd, Oct 14 2024
STATUS
approved