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A230708 T(n,k)=Number of (n+3)X(k+3) 0..2 white square subarrays x(i,j) with each element diagonally or antidiagonally next to at least one element with value (x(i,j)+1) mod 3, and upper left element zero.

%I A230708 #6 Jul 23 2025 05:58:11
%S A230708 2,8,8,30,28,30,102,244,244,102,348,800,2106,800,348,1172,6576,16536,
%T A230708 16536,6576,1172,3956,21076,130446,121382,130446,21076,3956,13326,
%U A230708 173428,1025430,2382398,2382398,1025430,173428,13326,44916,554040,8053490
%N A230708 T(n,k)=Number of (n+3)X(k+3) 0..2 white square subarrays x(i,j) with each element diagonally or antidiagonally next to at least one element with value (x(i,j)+1) mod 3, and upper left element zero.
%C A230708 Table starts
%C A230708 .....2......8.......30........102..........348...........1172.............3956
%C A230708 .....8.....28......244........800.........6576..........21076...........173428
%C A230708 ....30....244.....2106......16536.......130446........1025430..........8053490
%C A230708 ...102....800....16536.....121382......2382398.......17497342........342031378
%C A230708 ...348...6576...130446....2382398.....43853402......801845362......14669811856
%C A230708 ..1172..21076..1025430...17497342....801845362....13663471254.....625553036008
%C A230708 ..3956.173428..8053490..342031378..14669811856...625553036008...26681634560690
%C A230708 .13326.554040.63237238.2508344646.268320990890.10652997961328.1137681116923966
%H A230708 R. H. Hardin, <a href="/A230708/b230708.txt">Table of n, a(n) for n = 1..219</a>
%F A230708 Empirical for column k:
%F A230708 k=1: a(n) = 3*a(n-1) +3*a(n-2) -6*a(n-3) +a(n-5)
%F A230708 k=2: a(n) = 31*a(n-2) -126*a(n-4) +42*a(n-6) +79*a(n-8) -a(n-10) +8*a(n-12)
%F A230708 k=3: [order 22]
%F A230708 k=4: [order 50]
%e A230708 Some solutions for n=2 k=4
%e A230708 ..0..x..2..x..2..x..0....0..x..2..x..2..x..2....0..x..0..x..2..x..0
%e A230708 ..x..1..x..0..x..1..x....x..1..x..0..x..0..x....x..1..x..0..x..1..x
%e A230708 ..0..x..1..x..2..x..0....0..x..0..x..1..x..2....1..x..2..x..1..x..1
%e A230708 ..x..2..x..0..x..1..x....x..2..x..2..x..1..x....x..2..x..1..x..2..x
%e A230708 ..1..x..1..x..0..x..0....1..x..1..x..0..x..0....1..x..0..x..0..x..1
%K A230708 nonn,tabl
%O A230708 1,1
%A A230708 _R. H. Hardin_, Oct 28 2013