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Number of (n+2) X 8 binary arrays with each 3 X 3 subblock having rows and columns in lexicographically nondecreasing order.
1

%I #12 Apr 14 2018 02:37:12

%S 415,1114,2413,4774,8989,16345,28844,49489,82648,134509,213640,331669,

%T 504100,751282,1099549,1582550,2242789,3133396,4320151,5883784,

%U 7922575,10555279,13924402,18199855,23583014,30311215,38662714,48962143,61586494

%N Number of (n+2) X 8 binary arrays with each 3 X 3 subblock having rows and columns in lexicographically nondecreasing order.

%C Column 6 of A184548.

%H R. H. Hardin, <a href="/A184545/b184545.txt">Table of n, a(n) for n = 1..200</a>

%F Empirical: a(n) = (1/40320)*n^8 + (13/10080)*n^7 + (83/2880)*n^6 + (13/36)*n^5 + (15967/5760)*n^4 + (19201/1440)*n^3 + (121183/1120)*n^2 + (12673/56)*n + 64.

%F Conjectures from _Colin Barker_, Apr 13 2018: (Start)

%F G.f.: x*(415 - 2621*x + 7327*x^2 - 11699*x^3 + 11605*x^4 - 7310*x^5 + 2861*x^6 - 641*x^7 + 64*x^8) / (1 - x)^9.

%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>9.

%F (End)

%e Some solutions for 4 X 8:

%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

%e 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

%e 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0

%e 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1

%Y Cf. A184548.

%K nonn

%O 1,1

%A _R. H. Hardin_, Jan 16 2011