A250755 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing x(i,j)+x(i-1,j) in the j direction

32, 72, 105, 129, 237, 332, 203, 423, 756, 1029, 294, 663, 1353, 2361, 3152, 402, 957, 2123, 4239, 7272, 9585, 527, 1305, 3066, 6663, 13089, 22197, 29012, 669, 1707, 4182, 9633, 20603, 40023, 67356, 87549, 828, 2163, 5471, 13149, 29814, 63063, 121593

Offset

1,1

Comments

Table starts

.....32......72.....129.....203.....294......402......527......669......828

....105.....237.....423.....663.....957.....1305.....1707.....2163.....2673

....332.....756....1353....2123....3066.....4182.....5471.....6933.....8568

...1029....2361....4239....6663....9633....13149....17211....21819....26973

...3152....7272...13089...20603...29814....40722....53327....67629....83628

...9585...22197...40023...63063...91317...124785...163467...207363...256473

..29012...67356..121593..191723..277746...379662...497471...631173...780768

..87549..203601..367839..580263..840873..1149669..1506651..1911819..2365173

.263672..613872.1109649.1751003.2537934..3470442..4548527..5772189..7141428

.793065.1847757.3341223.5273463.7644477.10454265.13702827.17390163.21516273

Links

R. H. Hardin, Table of n, a(n) for n = 1..241

Formula

Empirical: T(n,k) = (3*(k+1)*(5*k+4)*3^n - (8*k^2+8*k)*2^n + (5*k^2-7*k))/4

Empirical for column k:

k=1: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (27*3^n-8*2^n-1)/2

k=2: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (63*3^n-24*2^n+3)/2

k=3: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (114*3^n-48*2^n+12)/2

k=4: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (180*3^n-80*2^n+26)/2

k=5: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (261*3^n-120*2^n+45)/2

k=6: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (357*3^n-168*2^n+69)/2

k=7: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (468*3^n-224*2^n+98)/2

Empirical for row n:

n=1: a(n) = (17/2)*n^2 + (29/2)*n + 9

n=2: a(n) = 27*n^2 + 51*n + 27

n=3: a(n) = (173/2)*n^2 + (329/2)*n + 81

n=4: a(n) = 273*n^2 + 513*n + 243

n=5: a(n) = (1697/2)*n^2 + (3149/2)*n + 729

n=6: a(n) = 2607*n^2 + 4791*n + 2187

n=7: a(n) = (15893/2)*n^2 + (29009/2)*n + 6561

Example

Some solutions for n=4 k=4
..1..1..1..1..0....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
..1..1..1..1..2....2..2..2..2..2....1..1..1..1..1....0..0..0..0..0
..1..1..1..1..2....0..0..0..0..0....0..0..0..0..0....1..1..1..1..1
..0..1..1..1..2....1..1..1..2..2....2..2..2..2..2....0..0..2..2..2
..0..1..1..1..2....0..0..0..1..2....0..1..1..1..2....0..0..2..2..2

Crossrefs

Column 1 is A053152(n+3)

Sequence in context: A228665 A101539 A106700 * A216427 A104026 A216417

Adjacent sequences: A250752 A250753 A250754 * A250756 A250757 A250758

Keyword

nonn,tabl

Author

R. H. Hardin, Nov 27 2014

Status

approved