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.

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

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: A145009 A352352 A090229 * A204713 A241985 A194408

Adjacent sequences: A259219 A259220 A259221 * A259223 A259224 A259225

Keyword

nonn,tabl

Author

R. H. Hardin, Jun 21 2015

Status

approved