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A188843 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally or vertically. 6
2, 4, 3, 8, 8, 4, 16, 21, 13, 5, 32, 55, 40, 19, 6, 64, 144, 121, 66, 26, 7, 128, 377, 364, 221, 100, 34, 8, 256, 987, 1093, 728, 364, 143, 43, 9, 512, 2584, 3280, 2380, 1288, 560, 196, 53, 10, 1024, 6765, 9841, 7753, 4488, 2108, 820, 260, 64, 11, 2048, 17711, 29524 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Table starts

   2  4   8   16   32    64    128    256     512     1024     2048      4096

   3  8  21   55  144   377    987   2584    6765    17711    46368    121393

   4 13  40  121  364  1093   3280   9841   29524    88573   265720    797161

   5 19  66  221  728  2380   7753  25213   81927   266110   864201   2806272

   6 26 100  364 1288  4488  15504  53296  182688   625184  2137408   7303360

   7 34 143  560 2108  7752  28101 100947  360526  1282735  4552624  16131656

   8 43 196  820 3264 12597  47652 177859  657800  2417416  8844448  32256553

   9 53 260 1156 4845 19551  76912 297275 1134705  4292145 16128061  60304951

  10 64 336 1581 6954 29260 119416 476905 1874730  7283640 28048800 107286661

  11 76 425 2109 9709 42504 179630 740025 2991495 11920740 46981740 183579396

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..1741

FORMULA

Row recurrence

Empirical: T(n,k) = Sum_{i=1..floor((n+2)/2)} binomial(n+2-i,i)*T(n,k-i)*(-1)^(i-1).

E.g.,

empirical: T(1,k) = 2*T(1,k-1),

empirical: T(2,k) = 3*T(2,k-1) -    T(2,k-2),

empirical: T(3,k) = 4*T(3,k-1) -  3*T(3,k-2),

empirical: T(4,k) = 5*T(4,k-1) -  6*T(4,k-2) +    T(4,k-3),

empirical: T(5,k) = 6*T(5,k-1) - 10*T(5,k-2) +  4*T(5,k-3),

empirical: T(6,k) = 7*T(6,k-1) - 15*T(6,k-2) + 10*T(6,k-3) -    T(6,k-4),

empirical: T(7,k) = 8*T(7,k-1) - 21*T(7,k-2) + 20*T(7,k-3) -  5*T(7,k-4),

empirical: T(8,k) = 9*T(8,k-1) - 28*T(8,k-2) + 35*T(8,k-3) - 15*T(8,k-4) + T(8,k-5).

Columns are polynomials for n > k-3.

Empirical: T(n,1) = n + 1.

Empirical: T(n,2) = (1/2)*n^2 + (5/2)*n + 1.

Empirical: T(n,3) = (1/6)*n^3 + 2*n^2 + (35/6)*n.

Empirical: T(n,4) = (1/24)*n^4 + (11/12)*n^3 + (155/24)*n^2 + (163/12)*n - 6 for n > 1.

Empirical: T(n,5) = (1/120)*n^5 + (7/24)*n^4 + (89/24)*n^3 + (473/24)*n^2 + (1877/60)*n - 33 for n > 2.

Empirical: T(n,6) = (1/720)*n^6 + (17/240)*n^5 + (203/144)*n^4 + (647/48)*n^3 + (2659/45)*n^2 + (1379/20)*n - 143 for n > 3.

Empirical: T(n,7) = (1/5040)*n^7 + (1/72)*n^6 + (143/360)*n^5 + (53/9)*n^4 + (33667/720)*n^3 + (12679/72)*n^2 + (9439/70)*n - 572 for n > 4.

Empirical: T(n,8) = (1/40320)*n^8 + (23/10080)*n^7 + (17/192)*n^6 + (269/144)*n^5 + (43949/1920)*n^4 + (228401/1440)*n^3 + (1054411/2016)*n^2 + (9941/56)*n - 2210 for n > 5.

EXAMPLE

Some solutions for 5 X 3:

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

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

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

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

  0 0 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0

CROSSREFS

Diagonal is A143388.

Column 2 is A034856(n+1).

Column 3 is A137742(n+1).

Row 2 is A001906(n+1).

Row 3 is A003462(n+1).

Row 4 is A005021.

Row 5 is A005022.

Row 6 is A005023.

Row 7 is A005024.

Row 8 is A005025.

Sequence in context: A111699 A067179 A318993 * A209406 A188706 A304408

Adjacent sequences:  A188840 A188841 A188842 * A188844 A188845 A188846

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin, Apr 12 2011

STATUS

approved

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Last modified May 28 09:09 EDT 2022. Contains 354112 sequences. (Running on oeis4.)