A250769 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing absolute value of x(i,j)-x(i-1,j) in the j direction.

9, 18, 18, 35, 34, 36, 68, 62, 66, 72, 133, 114, 114, 130, 144, 262, 214, 196, 216, 258, 288, 519, 410, 344, 350, 418, 514, 576, 1032, 798, 622, 572, 648, 820, 1026, 1152, 2057, 1570, 1158, 962, 996, 1234, 1622, 2050, 2304, 4106, 3110, 2208, 1680, 1558, 1812

Offset: 1

Author

R. H. Hardin, Nov 27 2014

Keywords

• nonn
• tabl

Comments

Table starts

....9...18....35....68...133...262...519..1032..2057..4106...8203..16396..32781

...18...34....62...114...214...410...798..1570..3110..6186..12334..24626..49206

...36...66...114...196...344...622..1158..2208..4284..8410..16634..33052..65856

...72..130...216...350...572...962..1680..3046..5700.10922..21272..41870..82956

..144..258...418...648...996..1558..2526..4284..7600.14010..26586..51472.100956

..288..514...820..1234..1812..2666..4020..6322.10468.18250..33252..62642.120756

..576.1026..1622..2396..3412..4798..6810..9960.15272.24794..42622..76948.144156

.1152.2050..3224..4710..6580..8978.12192.16798.23948.35946..57400..97526.174756

.2304.4098..6426..9328.12884.17254.22758.30036.40368.56314..82994.130648.219756

.4608.8194.12828.18554.25460.33722.43692.56074.72276.95114.130220.188858.293556

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..611

Crossrefs

Column 1 is A005010(n-1)

Column 2 is A052548(n+3)

Row 1 is A083706(n+1)

Formula

Empirical for column k: (k+2)^2*2^(n-1) plus a linear polynomial in n

k=1: a(n) = 2*a(n-1); a(n) = 9*2^(n-1)

k=2: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 16*2^(n-1) + 2

k=3: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 25*2^(n-1) + 2*n + 8

k=4: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 36*2^(n-1) + 10*n + 22

k=5: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 49*2^(n-1) + 32*n + 52

k=6: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 64*2^(n-1) + 84*n + 114

k=7: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 81*2^(n-1) + 198*n + 240

k=8: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 100*2^(n-1) + 438*n + 494

k=9: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 121*2^(n-1) + 932*n + 1004

Empirical for row n: (4*n+4)*2^(k-1) plus a quadratic polynomial in k

n=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 8*2^(n-1) + n

n=2: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 12*2^(n-1) + 4*n + 2

n=3: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 16*2^(n-1) + n^2 + 11*n + 8

n=4: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 20*2^(n-1) + 4*n^2 + 26*n + 22

n=5: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 24*2^(n-1) + 11*n^2 + 57*n + 52

n=6: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 28*2^(n-1) + 26*n^2 + 120*n + 114

n=7: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 32*2^(n-1) + 57*n^2 + 247*n + 240

n=8: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 36*2^(n-1) + 120*n^2 + 502*n + 494

n=9: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 40*2^(n-1) + 247*n^2 + 1013*n + 1004