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

Author

R. H. Hardin Feb 15 2012

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

Crossrefs

Column 2 is A016742

Column 3 is A152746

Row 1 is A000930(n+3)

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