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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. 9
7, 13, 13, 24, 23, 24, 45, 40, 40, 45, 85, 71, 66, 71, 85, 162, 127, 112, 112, 127, 162, 311, 230, 192, 183, 192, 230, 311, 601, 421, 334, 303, 303, 334, 421, 601, 1168, 779, 588, 510, 487, 510, 588, 779, 1168, 2281, 1456, 1048, 869, 798, 798, 869, 1048, 1456, 2281 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Table starts

     7   13   24   45   85  162   311   601  1168  2281  4473   8802  17371

    13   23   40   71  127  230   421   779  1456  2747  5227  10022  19345

    24   40   66  112  192  334   588  1048  1890  3448  6360  11854  22308

    45   71  112  183  303  510   869  1499  2616  4619  8251  14910  27249

    85  127  192  303  487  798  1325  2227  3784  6499 11283  19806  35161

   162  230  334  510  798 1278  2078  3422  5694  9566 16222  27774  48030

   311  421  588  869 1325 2078  3319  5377  8804 14545 24225  40670  68843

   601  779 1048 1499 2227 3422  5377  8591 13888 22655 37231  61598 102589

  1168 1456 1890 2616 3784 5694  8804 13888 22210 35872 58368  95550 157276

  2281 2747 3448 4619 6499 9566 14545 22655 35872 57455 92767 150686 245965

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

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..480

FORMULA

Empirical for diagonal and column k (k=3..7 recurrences work also for k=1,2):

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).

k=1: a(n) = 3*a(n-1) -   a(n-2) - 2*a(n-3)

k=2: a(n) = 3*a(n-1) -   a(n-2) - 2*a(n-3)

k=3: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)

k=4: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)

k=5: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)

k=6: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)

k=7: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)

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

EXAMPLE

Some solutions for n=4, k=4:

  0 0 1 0 1      1 1 1 0 1      0 0 0 0 1      0 0 0 1 0

  0 0 1 0 1      0 0 0 1 0      0 0 0 0 1      0 0 0 1 0

  1 1 0 1 0      0 0 0 1 0      1 1 1 1 0      0 0 0 1 0

  0 0 1 0 1      0 0 0 1 0      0 0 0 0 1      1 1 1 0 1

  1 1 0 1 0      0 0 0 1 0      0 0 0 0 1      0 0 0 1 0

CROSSREFS

Cf. A259215, A259216, A259217, A259218, A259219, A259220, A259221.

Sequence in context: A229831 A145009 A090229 * A204713 A241985 A194408

Adjacent sequences:  A259219 A259220 A259221 * A259223 A259224 A259225

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin, Jun 21 2015

STATUS

approved

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Last modified October 20 03:04 EDT 2021. Contains 348099 sequences. (Running on oeis4.)