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A259222 T(n,k) is the number of (n+1) X (k+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

%I A259222 #17 Oct 09 2020 12:13:02
%S A259222 7,13,13,24,23,24,45,40,40,45,85,71,66,71,85,162,127,112,112,127,162,
%T A259222 311,230,192,183,192,230,311,601,421,334,303,303,334,421,601,1168,779,
%U A259222 588,510,487,510,588,779,1168,2281,1456,1048,869,798,798,869,1048,1456,2281
%N A259222 T(n,k) is the number of (n+1) X (k+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.
%C A259222 Table starts
%C A259222      7   13   24   45   85  162   311   601  1168  2281  4473   8802  17371
%C A259222     13   23   40   71  127  230   421   779  1456  2747  5227  10022  19345
%C A259222     24   40   66  112  192  334   588  1048  1890  3448  6360  11854  22308
%C A259222     45   71  112  183  303  510   869  1499  2616  4619  8251  14910  27249
%C A259222     85  127  192  303  487  798  1325  2227  3784  6499 11283  19806  35161
%C A259222    162  230  334  510  798 1278  2078  3422  5694  9566 16222  27774  48030
%C A259222    311  421  588  869 1325 2078  3319  5377  8804 14545 24225  40670  68843
%C A259222    601  779 1048 1499 2227 3422  5377  8591 13888 22655 37231  61598 102589
%C A259222   1168 1456 1890 2616 3784 5694  8804 13888 22210 35872 58368  95550 157276
%C A259222   2281 2747 3448 4619 6499 9566 14545 22655 35872 57455 92767 150686 245965
%C A259222 Each row (and each column, by symmetry) has a rational generating function (and therefore a linear recurrence with constant coefficients) because the growth from an array to the next larger one is described by the transfer matrix method. - _R. J. Mathar_, Oct 09 2020
%H A259222 R. H. Hardin, <a href="/A259222/b259222.txt">Table of n, a(n) for n = 1..480</a>
%F A259222 Empirical for diagonal and column k (k=3..7 recurrences work also for k=1,2):
%F A259222 diagonal: a(n) = 6*a(n-1) - 10*a(n-2) - 2*a(n-3) + 16*a(n-4) - 6*a(n-5) - 5*a(n-6) + 2*a(n-7).
%F A259222 k=1: a(n) = 3*a(n-1) -   a(n-2) - 2*a(n-3)
%F A259222 k=2: a(n) = 3*a(n-1) -   a(n-2) - 2*a(n-3)
%F A259222 k=3: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
%F A259222 k=4: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
%F A259222 k=5: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
%F A259222 k=6: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
%F A259222 k=7: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
%F A259222 Empirical: T(n,k) = 2^(k+1) + 2^(n+1) + F(n+3)*F(k+3) - 2*F(n+3) - 2*F(k+3) + 2 = 2^(n+1) + A001911(k)*F(n+3) + A234933(k+1) = A234933(n+1) + A234933(k+1) + A143211(n+3,k+3) - 2, F=A000045. - _Ehren Metcalfe_, Dec 27 2018
%e A259222 Some solutions for n=4, k=4:
%e A259222   0 0 1 0 1      1 1 1 0 1      0 0 0 0 1      0 0 0 1 0
%e A259222   0 0 1 0 1      0 0 0 1 0      0 0 0 0 1      0 0 0 1 0
%e A259222   1 1 0 1 0      0 0 0 1 0      1 1 1 1 0      0 0 0 1 0
%e A259222   0 0 1 0 1      0 0 0 1 0      0 0 0 0 1      1 1 1 0 1
%e A259222   1 1 0 1 0      0 0 0 1 0      0 0 0 0 1      0 0 0 1 0
%Y A259222 Cf. A259215, A259216, A259217, A259218, A259219, A259220, A259221.
%K A259222 nonn,tabl
%O A259222 1,1
%A A259222 _R. H. Hardin_, Jun 21 2015