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A202052 T(n,k)=Number of (n+2)X(k+2) binary arrays avoiding patterns 001 and 110 in rows and columns.

%I A202052 #7 Jul 22 2025 16:27:42
%S A202052 102,216,216,390,528,390,636,1080,1080,636,966,1968,2470,1968,966,
%T A202052 1392,3304,4980,4980,3304,1392,1926,5216,9170,11016,9170,5216,1926,
%U A202052 2580,7848,15760,22092,22092,15760,7848,2580,3366,11360,25650,41088,47950,41088
%N A202052 T(n,k)=Number of (n+2)X(k+2) binary arrays avoiding patterns 001 and 110 in rows and columns.
%C A202052 Table starts
%C A202052 ..102...216...390....636....966....1392....1926.....2580.....3366.....4296
%C A202052 ..216...528..1080...1968...3304....5216....7848....11360....15928....21744
%C A202052 ..390..1080..2470...4980...9170...15760...25650....39940....59950....87240
%C A202052 ..636..1968..4980..11016..22092...41088...71964...120000...192060...296880
%C A202052 ..966..3304..9170..22092..47950...95984..180054...320180...544390...890904
%C A202052 .1392..5216.15760..41088..95984..205792..411696...777760..1400080..2418432
%C A202052 .1926..7848.25650..71964.180054..411696..874998..1750140..3325410..6046344
%C A202052 .2580.11360.39940.120000.320180..777760.1750140..3694920..7390020.14108400
%C A202052 .3366.15928.59950.192060.544390.1400080.3325410..7390020.15519262.31038744
%C A202052 .4296.21744.87240.296880.890904.2418432.6046344.14108400.31038744.64899456
%H A202052 R. H. Hardin, <a href="/A202052/b202052.txt">Table of n, a(n) for n = 1..511</a>
%F A202052 Empirical (via A086113): T(n,k)=2*(n+2)*(2*binomial(n+k+3,n+2)-k-2)
%F A202052 Empirical for columns:
%F A202052 T(n,1) = 2*n^3 + 18*n^2 + 46*n + 36
%F A202052 T(n,2) = (2/3)*n^4 + (28/3)*n^3 + (142/3)*n^2 + (284/3)*n + 64
%F A202052 T(n,3) = (1/6)*n^5 + (10/3)*n^4 + (155/6)*n^3 + (290/3)*n^2 + 164*n + 100
%F A202052 T(n,4) = (1/30)*n^6 + (9/10)*n^5 + (59/6)*n^4 + (111/2)*n^3 + (2552/15)*n^2 + (1278/5)*n + 144
%F A202052 T(n,5) = (1/180)*n^7 + (7/36)*n^6 + (511/180)*n^5 + (805/36)*n^4 + (4606/45)*n^3 + (2443/9)*n^2 + (1854/5)*n + 196
%F A202052 T(n,6) = (1/1260)*n^8 + (11/315)*n^7 + (59/90)*n^6 + (308/45)*n^5 + (7807/180)*n^4 + (7667/45)*n^3 + (14139/35)*n^2 + (17876/35)*n + 256
%F A202052 T(n,7) = (1/10080)*n^9 + (3/560)*n^8 + (211/1680)*n^7 + (67/40)*n^6 + (6709/480)*n^5 + (6041/80)*n^4 + (663941/2520)*n^3 + (79913/140)*n^2 + (4735/7)*n + 324
%e A202052 Some solutions for n=5 k=3
%e A202052 ..0..0..0..0..0....1..0..1..1..1....0..1..0..1..1....0..1..0..1..1
%e A202052 ..1..1..1..1..1....0..1..0..0..0....1..0..1..0..0....1..0..0..0..0
%e A202052 ..0..0..0..0..0....1..0..1..1..1....0..1..0..1..0....0..1..0..1..1
%e A202052 ..0..1..1..1..1....0..0..0..0..0....0..0..0..0..0....1..0..0..0..0
%e A202052 ..0..1..0..0..0....1..0..1..1..1....0..1..0..1..0....0..1..0..1..1
%e A202052 ..0..1..0..1..1....0..0..0..0..0....0..1..0..1..0....0..0..0..0..0
%e A202052 ..0..1..0..1..0....1..0..1..0..0....0..1..0..1..0....0..1..0..1..0
%Y A202052 Column 1 is A086113(n+2)
%Y A202052 Column 2 is A086114(n+2)
%Y A202052 Column 3 is A086115(n+2)
%Y A202052 Diagonal is A032260(n+2)
%K A202052 nonn,tabl
%O A202052 1,1
%A A202052 _R. H. Hardin_ Dec 10 2011