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A224550 T(n,k)=Number of (n+1)X(k+1) 0..1 matrices with each 2X2 subblock idempotent.

%I A224550 #6 Jul 23 2025 05:10:16
%S A224550 8,16,16,28,32,28,48,52,52,48,80,86,78,86,80,132,137,125,125,137,132,
%T A224550 216,218,193,196,193,218,216,352,345,302,294,294,302,345,352,572,547,
%U A224550 472,448,428,448,472,547,572,928,869,743,682,635,635,682,743,869,928,1504,1385
%N A224550 T(n,k)=Number of (n+1)X(k+1) 0..1 matrices with each 2X2 subblock idempotent.
%C A224550 Table starts
%C A224550 ...8...16...28...48...80..132..216..352..572...928..1504..2436..3944..6384
%C A224550 ..16...32...52...86..137..218..345..547..869..1385..2214..3549..5702..9178
%C A224550 ..28...52...78..125..193..302..472..743.1175..1868..2982..4777..7673.12350
%C A224550 ..48...86..125..196..294..448..682.1049.1626..2543..4007..6355.10131.16216
%C A224550 ..80..137..193..294..428..635..943.1420.2162..3333..5193..8166.12936.20611
%C A224550 .132..218..302..448..635..916.1323.1941.2887..4363..6688.10383.16288.25764
%C A224550 .216..345..472..682..943.1323.1858.2652.3845..5681..8544.13062.20247.31739
%C A224550 .352..547..743.1049.1420.1941.2652.3678.5184..7460.10961.16432.25073.38829
%C A224550 .572..869.1175.1626.2162.2887.3845.5184.7096..9921.14191.20778.31086.47391
%C A224550 .928.1385.1868.2543.3333.4363.5681.7460.9921.13458.18686.26614.38866.58075
%C A224550 Acceptable 2X2 patterns
%C A224550 ..1..0....1..1....0..0....0..0....1..0....1..0....0..1....0..0
%C A224550 ..0..1....0..0....0..1....1..1....1..0....0..0....0..1....0..0
%C A224550 Forbidden 2X2 patterns
%C A224550 ..1..1....0..1....1..0....0..1....1..1....0..0....1..1....0..1
%C A224550 ..1..0....1..0....1..1....1..1....1..1....1..0....0..1....0..0
%H A224550 R. H. Hardin, <a href="/A224550/b224550.txt">Table of n, a(n) for n = 1..1103</a>
%F A224550 Empirical for diagonal:
%F A224550 a(n) = 6*a(n-1) -12*a(n-2) +5*a(n-3) +12*a(n-4) -12*a(n-5) -3*a(n-6) +6*a(n-7) -a(n-9)
%F A224550 Empirical for column k:
%F A224550 k=1: a(n) = 2*a(n-1) -a(n-3)
%F A224550 k=2: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5)
%F A224550 k=3..12+: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5) for n>6
%e A224550 Some solutions for n=3 k=4
%e A224550 ..1..1..0..0..0....1..0..1..0..1....1..0..0..0..0....1..0..1..0..0
%e A224550 ..0..0..0..0..0....1..0..1..0..1....1..0..0..0..0....1..0..1..0..0
%e A224550 ..1..1..1..1..1....0..0..1..0..1....0..0..0..0..0....1..0..1..0..0
%e A224550 ..0..0..0..0..0....0..0..1..0..1....0..0..0..1..1....0..0..1..0..1
%Y A224550 Column 1 is A204644
%K A224550 nonn,tabl
%O A224550 1,1
%A A224550 _R. H. Hardin_ Apr 10 2013