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%I #11 Mar 09 2024 16:25:50
%S 4,6,6,9,8,9,15,11,11,15,25,17,14,17,25,45,27,20,20,27,45,81,47,30,26,
%T 30,47,81,153,83,50,36,36,50,83,153,289,155,86,56,46,56,86,155,289,
%U 561,291,158,92,66,66,92,158,291,561,1089,563,294,164,102,86,102,164,294,563
%N T(n,k) = 1/4 the number of (n+1) X (k+1) binary arrays with all 2 X 2 subblock sums the same.
%C Table starts
%C ...4...6...9..15..25..45..81.153.289..561.1089.2145.4225.8385.16641.33153.66049
%C ...6...8..11..17..27..47..83.155.291..563.1091.2147.4227.8387.16643.33155.66051
%C ...9..11..14..20..30..50..86.158.294..566.1094.2150.4230.8390.16646.33158.66054
%C ..15..17..20..26..36..56..92.164.300..572.1100.2156.4236.8396.16652.33164.66060
%C ..25..27..30..36..46..66.102.174.310..582.1110.2166.4246.8406.16662.33174.66070
%C ..45..47..50..56..66..86.122.194.330..602.1130.2186.4266.8426.16682.33194.66090
%C ..81..83..86..92.102.122.158.230.366..638.1166.2222.4302.8462.16718.33230.66126
%C .153.155.158.164.174.194.230.302.438..710.1238.2294.4374.8534.16790.33302.66198
%C .289.291.294.300.310.330.366.438.574..846.1374.2430.4510.8670.16926.33438.66334
%C .561.563.566.572.582.602.638.710.846.1118.1646.2702.4782.8942.17198.33710.66606
%H R. H. Hardin, <a href="/A183986/b183986.txt">Table of n, a(n) for n = 1..683</a>
%F Empirical, for every row and column: a(n) = 3*a(n-1)-6*a(n-3)+4*a(n-4).
%F From _Andrew Howroyd_, Mar 09 2024: (Start)
%F The above empirical formula is correct.
%F T(n,k) = -2 + 2^(n-1) + 2^(k-1) + 2^(floor((n-1)/2)) + 2^(floor(n/2)) + 2^(floor((k-1)/2)) + 2^(floor(k/2)). (End)
%e Some solutions for 6 X 5
%e ..0..1..0..0..1....1..1..1..1..1....1..1..0..0..1....1..1..0..1..0
%e ..1..0..1..1..0....1..0..1..0..1....0..0..1..1..0....0..0..1..0..1
%e ..0..1..0..0..1....1..1..1..1..1....1..1..0..0..1....1..1..0..1..0
%e ..1..0..1..1..0....0..1..0..1..0....0..0..1..1..0....0..0..1..0..1
%e ..0..1..0..0..1....1..1..1..1..1....1..1..0..0..1....1..1..0..1..0
%e ..1..0..1..1..0....0..1..0..1..0....0..0..1..1..0....0..0..1..0..1
%o (PARI) T(n,k) = my(m=2, b=t->2^t-1); m^2 + (m-1)^2*(b(n-1) + b(k-1)) + (m-1)*(b((n-1)\2) + b(n\2) + b((k-1)\2) + b(k\2)) \\ _Andrew Howroyd_, Mar 09 2024
%Y Columns 1..8 are A183978, A183979, A183980, A183981, A183982, A183983, A183984, A183985.
%Y Main diagonal is A183977.
%Y Cf. A184039, A184048.
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Jan 08 2011
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