A207169 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 0 1 and 1 0 1 vertically

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 13, 81, 90, 64, 10, 19, 169, 261, 168, 100, 12, 28, 361, 624, 603, 270, 144, 14, 41, 784, 1482, 1612, 1161, 396, 196, 16, 60, 1681, 3808, 3952, 3445, 1989, 546, 256, 18, 88, 3600, 9512, 11452, 8455, 6513, 3141, 720, 324, 20, 129, 7744

Offset

1,1

Comments

Table starts

..2...4...6....9....13....19.....28.....41......60......88......129......189

..4..16..36...81...169...361....784...1681....3600....7744....16641....35721

..6..36..90..261...624..1482...3808...9512...23280...58080...144996...359100

..8..64.168..603..1612..3952..11452..32021...84300..231616...641775..1736910

.10.100.270.1161..3445..8455..26908..82861..228060..672760..2029041..5846337

.12.144.396.1989..6513.15789..54208.182081..515760.1608288..5222049.15774129

.14.196.546.3141.11284.26866..98224.357356.1032000.3365824.11680176.36617616

.16.256.720.4671.18304.42712.164668.645217.1888380.6392320.23581071.76187790

Links

R. H. Hardin, Table of n, a(n) for n = 1..1512

Formula

Empirical for column k:

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

k=2: a(n) = 4*n^2

k=3: a(n) = 12*n^2 - 6*n

k=4: a(n) = 9*n^3 + 9*n - 9

k=5: a(n) = (13/4)*n^4 + (13/2)*n^3 + (117/4)*n^2 - 26*n

k=6: a(n) = (19/4)*n^4 + (95/2)*n^3 - (57/4)*n^2 - 19*n

k=7: a(n) = 35*n^4 + 42*n^3 + 7*n^2 - 84*n + 28

Empirical for rows:

n=1: a(k)=a(k-1)+a(k-3) for k>4

n=2: a(k)=a(k-1)+a(k-2)+3*a(k-3)+a(k-4)-a(k-5)-a(k-6) for k>7

n=3: a(k)=a(k-1)+9*a(k-3)+2*a(k-4)+2*a(k-5)-12*a(k-6)-8*a(k-7)+8*a(k-9) for k>11

n=4: a(k)=a(k-1)+13*a(k-3)+3*a(k-4)+3*a(k-5)-27*a(k-6)-18*a(k-7)+27*a(k-9) for k>11

n=5: a(k)=a(k-1)+17*a(k-3)+4*a(k-4)+4*a(k-5)-48*a(k-6)-32*a(k-7)+64*a(k-9) for k>11

n=6: a(k)=a(k-1)+21*a(k-3)+5*a(k-4)+5*a(k-5)-75*a(k-6)-50*a(k-7)+125*a(k-9) for k>11

n=7: a(k)=a(k-1)+25*a(k-3)+6*a(k-4)+6*a(k-5)-108*a(k-6)-72*a(k-7)+216*a(k-9) for k>11

n=8: a(k)=a(k-1)+29*a(k-3)+7*a(k-4)+7*a(k-5)-147*a(k-6)-98*a(k-7)+343*a(k-9) for k>11

n=9: a(k)=a(k-1)+33*a(k-3)+8*a(k-4)+8*a(k-5)-192*a(k-6)-128*a(k-7)+512*a(k-9) for k>11

n=10: a(k)=a(k-1)+37*a(k-3)+9*a(k-4)+9*a(k-5)-243*a(k-6)-162*a(k-7)+729*a(k-9) for k>11

n=11: a(k)=a(k-1)+41*a(k-3)+10*a(k-4)+10*a(k-5)-300*a(k-6)-200*a(k-7)+1000*a(k-9) for k>11

n=12: a(k)=a(k-1)+45*a(k-3)+11*a(k-4)+11*a(k-5)-363*a(k-6)-242*a(k-7)+1331*a(k-9) for k>11

n=13: a(k)=a(k-1)+49*a(k-3)+12*a(k-4)+12*a(k-5)-432*a(k-6)-288*a(k-7)+1728*a(k-9) for k>11

n=14: a(k)=a(k-1)+53*a(k-3)+13*a(k-4)+13*a(k-5)-507*a(k-6)-338*a(k-7)+2197*a(k-9) for k>11

n=15: a(k)=a(k-1)+57*a(k-3)+14*a(k-4)+14*a(k-5)-588*a(k-6)-392*a(k-7)+2744*a(k-9) for k>11

apparently a(k)=a(k-1)+(4*n-3)*a(k-3)+(n-1)*a(k-4)+(n-1)*a(k-5)-3*(n-1)^2*a(k-6)-2*(n-1)^2*a(k-7)+(n-1)^3*a(k-9) for n>2 and k>11

Example

Some solutions for n=4 k=3
..1..0..0....0..0..1....0..1..1....1..1..1....0..0..1....1..0..0....1..0..0
..1..0..0....0..1..1....0..0..1....1..1..1....0..0..1....1..1..0....0..0..1
..1..0..0....0..1..0....0..0..1....1..1..1....0..0..1....0..1..0....0..0..1
..1..0..0....0..1..0....0..0..1....1..1..1....0..0..1....0..1..0....0..0..1

Crossrefs

Column 2 is A016742

Column 3 is A152746

Row 1 is A000930(n+3)

Sequence in context: A207403 A208142 A207024 * A207111 A207305 A207391

Adjacent sequences: A207166 A207167 A207168 * A207170 A207171 A207172

Keyword

nonn,tabl

Author

R. H. Hardin Feb 15 2012

Status

approved