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A188843
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T(n,k)=Number of nXk binary arrays without the pattern 0 1 diagonally or vertically
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6
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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R. H. Hardin, Table of n, a(n) for n = 1..1741
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FORMULA
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Row recurrence
Empirical: T(n,k) = sum(binomial(n+2-i,i)*T(n,k-i)*(-1)^(i-1) , i=1..floor((n+2)/2))
eg.,
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
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EXAMPLE
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Some solutions for 5X3
..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
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CROSSREFS
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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: A166133 A111699 A067179 * A209406 A188706 A048767
Adjacent sequences: A188840 A188841 A188842 * A188844 A188845 A188846
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KEYWORD
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nonn,tabl
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AUTHOR
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R. H. Hardin Apr 12 2011
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STATUS
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approved
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