A202786 - OEIS

%I #16 Oct 14 2024 14:31:07

%S 1,140,5673,89520,790425,4756140,21841937,82112704,264639729,

%T 754898668,1950230969,4641494832,10309971465,21592075596,42980713761,

%U 81851507456,149924818657,265300850124,455235310153,759857498672,1237071456633,1968924291180,3069774212913,4696644466368

%N Number of 4 X 4 0..n arrays with row and column sums equal.

%C From _Robert Israel_, May 03 2019: (Start)

%C 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.

%C 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]).

%C Therefore a(n) should be quasi-polynomial in n. (End)

%H Andrew Howroyd, <a href="/A202786/b202786.txt">Table of n, a(n) for n = 0..40</a> (terms 1..21 from R. H. Hardin)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ehrhart_polynomial#Ehrhart_quasi-polynomials">Ehrhart quasi-polynomials</a>

%F 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

%e Some solutions for n=3

%e ..2..2..1..3....2..1..2..1....3..2..1..0....1..3..2..2....0..3..2..2

%e ..1..2..3..2....1..3..2..0....0..1..2..3....2..2..2..2....1..2..3..1

%e ..3..1..3..1....2..1..1..2....1..1..1..3....2..1..2..3....3..2..1..1

%e ..2..3..1..2....1..1..1..3....2..2..2..0....3..2..2..1....3..0..1..3

%Y Row n=4 of A202784.

%K nonn

%O 0,2

%A _R. H. Hardin_, Dec 24 2011

%E a(0)=1 prepended by _Andrew Howroyd_, Oct 14 2024