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A205193 T(n,k) = Number of (n+1) X (k+1) 0..1 arrays with the number of clockwise edge increases in every 2 X 2 subblock differing from each horizontal or vertical neighbor.

%I A205193 #7 Nov 12 2022 19:52:57
%S A205193 16,24,24,48,40,48,72,64,64,72,144,104,124,104,144,216,168,160,160,
%T A205193 168,216,432,272,292,256,292,272,432,648,440,384,384,384,384,440,648,
%U A205193 1296,712,708,576,736,576,708,712,1296,1944,1152,928,864,896,896,864,928,1152
%N A205193 T(n,k) = Number of (n+1) X (k+1) 0..1 arrays with the number of clockwise edge increases in every 2 X 2 subblock differing from each horizontal or vertical neighbor.
%C A205193 Table starts
%C A205193 ..16..24..48...72..144..216..432..648..1296..1944..3888..5832.11664.17496.34992
%C A205193 ..24..40..64..104..168..272..440..712..1152..1864..3016..4880..7896.12776.20672
%C A205193 ..48..64.124..160..292..384..708..928..1708..2240..4124..5408..9956.13056.24036
%C A205193 ..72.104.160..256..384..576..864.1312..1984..3008..4544..6880.10400.15744.23808
%C A205193 .144.168.292..384..736..896.1568.1920..3392..4224..7520..9344.16608.20608.36608
%C A205193 .216.272.384..576..896.1408.2048.2944..4224..6144..8960.13184.19328.28416.41600
%C A205193 .432.440.708..864.1568.2048.3904.4608..7872..9216.15808.18944.32960.39936.69824
%C A205193 .648.712.928.1312.1920.2944.4608.7168.10240.14336.19968.28160.39936.57344.82432
%H A205193 R. H. Hardin, <a href="/A205193/b205193.txt">Table of n, a(n) for n = 1..1300</a>
%F A205193 Empirical for column k:
%F A205193 k=1: a(n) = 3*a(n-2);
%F A205193 k=2: a(n) = a(n-1) +a(n-2);
%F A205193 k=3: a(n) = 2*a(n-2) +a(n-4) for n>5;
%F A205193 k=4: a(n) = a(n-1) +a(n-2) -a(n-3) +a(n-4) for n>6;
%F A205193 k=5: a(n) = 2*a(n-2) +a(n-6) for n>9;
%F A205193 k=6: a(n) = a(n-1) +a(n-2) -a(n-3) +a(n-4) -a(n-5) +a(n-6) for n>10;
%F A205193 k=7: a(n) = 2*a(n-2) +a(n-8) for n>13;
%F A205193 k=8: a(n) = a(n-1) +a(n-2) -a(n-3) +a(n-4) -a(n-5) +a(n-6) -a(n-7) +a(n-8) for n>14;
%F A205193 k=9: a(n) = 2*a(n-2) +a(n-10) for n>17;
%F A205193 k=10: a(n) = a(n-1) +a(n-2) -a(n-3) +a(n-4) -a(n-5) +a(n-6) -a(n-7) +a(n-8) -a(n-9) +a(n-10) for n>18;
%F A205193 k=11: a(n) = 2*a(n-2) +a(n-12) for n>21;
%F A205193 k=12: a(n) = a(n-1) +a(n-2) -a(n-3) +a(n-4) -a(n-5) +a(n-6) -a(n-7) +a(n-8) -a(n-9) +a(n-10) -a(n-11) +a(n-12) for n>22;
%F A205193 k=13: a(n) = 2*a(n-2) +a(n-14) for n>25;
%F A205193 k=14: a(n) = a(n-1) +a(n-2) -a(n-3) +a(n-4) -a(n-5) +a(n-6) -a(n-7) +a(n-8) -a(n-9) +a(n-10) -a(n-11) +a(n-12) -a(n-13) +a(n-14) for n>26;
%F A205193 k=15: a(n) = 2*a(n-2) +a(n-16) for n>29;
%F A205193 apparently:
%F A205193 k odd a(n) = 2*a(n-2) +a(n-k-1) for n>2k-1;
%F A205193 k even a(n) = a(n-1) +sum{i in 2..k}(-1^i*a(n-i)) for n>2k-2.
%e A205193 Some solutions for n=4, k=3
%e A205193 ..1..0..0..1....0..1..1..0....1..1..1..0....0..0..1..1....1..1..0..0
%e A205193 ..0..1..1..0....0..1..1..1....0..1..1..0....0..0..1..1....1..1..0..0
%e A205193 ..1..1..1..0....1..0..1..1....1..0..0..1....1..1..0..0....0..0..1..1
%e A205193 ..1..1..0..1....1..1..0..1....0..0..0..1....1..1..0..0....0..0..1..1
%e A205193 ..1..0..1..1....1..1..1..0....0..0..1..0....0..0..1..0....1..1..0..1
%Y A205193 Column 2 is A022091(n+3).
%K A205193 nonn,tabl
%O A205193 1,1
%A A205193 _R. H. Hardin_, Jan 23 2012