login
Number of (n+1) X (n+1) -4..4 symmetric matrices with every 2 X 2 subblock having sum zero and three distinct values.
1

%I #7 Jul 18 2018 08:22:34

%S 28,56,110,206,392,728,1372,2562,4838,9126,17360,33116,63572,122558,

%T 237382,461802,901448,1766496,3470060,6838226,13498502,26712038,

%U 52921792,105046092,208674308,415112766,826217510,1646150186,3280944632

%N Number of (n+1) X (n+1) -4..4 symmetric matrices with every 2 X 2 subblock having sum zero and three distinct values.

%C Symmetry and 2 X 2 block sums zero implies that the diagonal x(i,i) are equal modulo 2 and x(i,j) = (x(i,i)+x(j,j))/2*(-1)^(i-j).

%H R. H. Hardin, <a href="/A211491/b211491.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 4*a(n-1) + a(n-2) - 21*a(n-3) + 16*a(n-4) + 29*a(n-5) - 34*a(n-6) - 6*a(n-7) + 12*a(n-8).

%F Empirical g.f.: 2*x*(14 - 28*x - 71*x^2 + 149*x^3 + 93*x^4 - 222*x^5 - 19*x^6 + 82*x^7) / ((1 - x)*(1 - 2*x)*(1 - x - x^2)*(1 - 2*x^2)*(1 - 3*x^2)). - _Colin Barker_, Jul 18 2018

%e Some solutions for n=3:

%e ..4.-2..1.-2...-2..3.-2..3...-2..1..0.-1....4.-1..4.-1....0.-1..0.-1

%e .-2..0..1..0....3.-4..3.-4....1..0.-1..2...-1.-2.-1.-2...-1..2.-1..2

%e ..1..1.-2..1...-2..3.-2..3....0.-1..2.-3....4.-1..4.-1....0.-1..0.-1

%e .-2..0..1..0....3.-4..3.-4...-1..2.-3..4...-1.-2.-1.-2...-1..2.-1..2

%K nonn

%O 1,1

%A _R. H. Hardin_, Apr 13 2012