A250877 T(n,k)=Number of (n+1)X(k+1) 0..3 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

90, 288, 440, 678, 1456, 2002, 1328, 3442, 6812, 8736, 2306, 6728, 16262, 30360, 37130, 3680, 11644, 31928, 73122, 131068, 155080, 5518, 18520, 55386, 144248, 317878, 553736, 640002, 7888, 27686, 88212, 250964, 629528, 1350002, 2304492

Offset

1,1

Comments

Table starts

.......90.......288.......678......1328.......2306.......3680.......5518

......440......1456......3442......6728......11644......18520......27686

.....2002......6812.....16262.....31928......55386......88212.....131982

.....8736.....30360.....73122....144248.....250964.....400496.....600070

....37130....131068....317878....629528....1097986....1755220....2633198

...155080....553736...1350002...2681528....4685964....7500960...11264166

...640002...2304492...5640102..11227928...19649066...31484612...47315662

..2619056...9488920..23289922..46440248...81358084..130461616..196169030

.10653370..38773148..95366678.190392728..333810066..535577460..805653678

.43144920.157554216.388124562.775558328.1360557884.2183825600.3286063846

Links

R. H. Hardin, Table of n, a(n) for n = 1..160

Formula

Empirical: T(n,k) = (((62/3)*k^3+52*k^2+(130/3)*k+12)*4^n -((45/2)*k^3+(81/2)*k^2+18*k)*3^n +(9*k^3-9*k)*2^n +((5/6)*k^3-(5/2)*k^2+(8/3)*k))/3

Empirical for column k:

k=1: a(n) = 8*a(n-1) -19*a(n-2) +12*a(n-3); a(n) = (128*4^n-81*3^n+1)/3

k=2: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (472*4^n-378*3^n+54*2^n+2)/3

k=3: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (1168*4^n-1026*3^n+216*2^n+8)/3

k=4: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (2340*4^n-2160*3^n+540*2^n+24)/3

k=5: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (4112*4^n-3915*3^n+1080*2^n+55)/3

k=6: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (6608*4^n-6426*3^n+1890*2^n+106)/3

k=7: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (9952*4^n-9828*3^n+3024*2^n+182)/3

Empirical for row n:

n=1: a(n) = (34/3)*n^3 + 28*n^2 + (104/3)*n + 16

n=2: a(n) = 55*n^3 + 155*n^2 + 166*n + 64

n=3: a(n) = (788/3)*n^3 + 744*n^2 + (2218/3)*n + 256

n=4: a(n) = (3613/3)*n^3 + 3343*n^2 + (9494/3)*n + 1024

n=5: a(n) = 5328*n^3 + 14468*n^2 + 13238*n + 4096

n=6: a(n) = (68825/3)*n^3 + 61155*n^2 + (163798/3)*n + 16384

n=7: a(n) = (290548/3)*n^3 + 254464*n^2 + (669458/3)*n + 65536

Example

Some solutions for n=4 k=4
..0..0..0..0..1....0..0..0..1..1....0..0..0..0..0....1..1..1..1..1
..2..2..2..2..3....2..2..2..3..3....2..2..3..3..3....0..0..0..0..0
..2..2..2..2..3....1..1..2..3..3....1..1..2..3..3....2..2..2..2..2
..0..1..1..1..2....0..0..1..2..2....1..1..2..3..3....0..2..2..2..2
..0..1..1..1..3....0..0..2..3..3....0..0..1..3..3....0..2..2..3..3

Crossrefs

Sequence in context: A237131 A363729 A255784 * A250878 A027621 A331259

Adjacent sequences: A250874 A250875 A250876 * A250878 A250879 A250880

Keyword

nonn,tabl

Author

R. H. Hardin, Nov 28 2014

Status

approved