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%I #4 Dec 31 2014 20:22:16
%S 16,39,39,69,58,69,109,70,73,109,181,102,85,108,181,325,174,120,121,
%T 180,325,613,318,192,156,193,324,613,1189,606,336,228,228,337,612,
%U 1189,2341,1182,624,372,300,372,625,1188,2341,4645,2334,1200,660,444,444,660,1201
%N T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with every 2X2 subblock diagonal minus antidiagonal sum nondecreasing horizontally, vertically and ne-to-sw antidiagonally
%C Table starts
%C ...16...39...69..109..181..325..613.1189.2341.4645..9253.18469.36901.73765
%C ...39...58...70..102..174..318..606.1182.2334.4638..9246.18462.36894.73758
%C ...69...73...85..120..192..336..624.1200.2352.4656..9264.18480.36912.73776
%C ..109..108..121..156..228..372..660.1236.2388.4692..9300.18516.36948.73812
%C ..181..180..193..228..300..444..732.1308.2460.4764..9372.18588.37020.73884
%C ..325..324..337..372..444..588..876.1452.2604.4908..9516.18732.37164.74028
%C ..613..612..625..660..732..876.1164.1740.2892.5196..9804.19020.37452.74316
%C .1189.1188.1201.1236.1308.1452.1740.2316.3468.5772.10380.19596.38028.74892
%C .2341.2340.2353.2388.2460.2604.2892.3468.4620.6924.11532.20748.39180.76044
%C .4645.4644.4657.4692.4764.4908.5196.5772.6924.9228.13836.23052.41484.78348
%H R. H. Hardin, <a href="/A253435/b253435.txt">Table of n, a(n) for n = 1..1101</a>
%F Empirical for diagonal:
%F diagonal: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F Empirical for column k:
%F k=1: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F k=2: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F k=3: a(n) = 3*a(n-1) -2*a(n-2) for n>4
%F k=4: a(n) = 3*a(n-1) -2*a(n-2) for n>3
%F k=5: a(n) = 3*a(n-1) -2*a(n-2) for n>3
%F k=6: a(n) = 3*a(n-1) -2*a(n-2) for n>3
%F k=7: a(n) = 3*a(n-1) -2*a(n-2) for n>3
%F Empirical for row n:
%F n=1: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F n=2: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F n=3: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F n=4: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F n=5: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F n=6: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F n=7: a(n) = 3*a(n-1) -2*a(n-2) for n>5
%F Empirical for diagonal:
%F diagonal: 9*2^n + 12 for n>3
%F Empirical for columns:
%F k=1: 9*2^(n-1) + 37 for n>3
%F k=2: 9*2^(n-1) + 36 for n>3
%F k=3: 9*2^(n-1) + 49 for n>2
%F k=4: 9*2^(n-1) + 84 for n>1
%F k=5: 9*2^(n-1) + 156 for n>1
%F k=6: 9*2^(n-1) + 300 for n>1
%F k=7: 9*2^(n-1) + 588 for n>1
%F Empirical for rows:
%F n=1: 9*2^(k-1) + 37 for k>3
%F n=2: 9*2^(k-1) + 30 for k>3
%F n=3: 9*2^(k-1) + 48 for k>3
%F n=4: 9*2^(k-1) + 84 for k>3
%F n=5: 9*2^(k-1) + 156 for k>3
%F n=6: 9*2^(k-1) + 300 for k>3
%F n=7: 9*2^(k-1) + 588 for k>3
%F Empirical: T(n,k) = 9*2(n-1) + 9*2^(k-1) + c, where
%F rows
%F n=1 c=28 for k>3
%F n>1 c=12 for k>3
%F columns
%F k=1 c=28 for n>3
%F k=2 c=18 for n>3
%F k=3 c=13 for n>2
%F k>3 c=12 for n>1
%F summary table of c
%F ..-2..12..24..28..28..28..28..28..28..28
%F ..12..22..16..12..12..12..12..12..12..12
%F ..24..19..13..12..12..12..12..12..12..12
%F ..28..18..13..12..12..12..12..12..12..12
%F ..28..18..13..12..12..12..12..12..12..12
%F ..28..18..13..12..12..12..12..12..12..12
%F ..28..18..13..12..12..12..12..12..12..12
%F ..28..18..13..12..12..12..12..12..12..12
%F ..28..18..13..12..12..12..12..12..12..12
%F ..28..18..13..12..12..12..12..12..12..12
%e Some solutions for n=4 k=4
%e ..1..0..0..0..1....1..1..1..1..1....1..1..1..0..0....0..0..0..0..0
%e ..1..0..0..0..1....1..1..1..1..1....1..1..1..0..0....1..1..1..1..1
%e ..1..0..0..0..1....1..1..1..1..1....1..1..1..0..0....1..1..1..1..1
%e ..1..0..0..0..1....0..0..0..0..0....1..1..1..0..0....0..0..0..0..0
%e ..1..0..0..0..1....0..0..0..0..0....1..1..1..0..0....1..1..1..1..1
%Y Column 1 and row 1 are A253152
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Dec 31 2014