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

1,1

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

Links

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

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

Example

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

Crossrefs

Column 1 is A005010(n-1)

Column 2 is A052548(n+3)

Row 1 is A083706(n+1)

Sequence in context: A040072 A034728 A129855 * A158908 A202188 A352960

Adjacent sequences: A250766 A250767 A250768 * A250770 A250771 A250772

Keyword

nonn,tabl

Author

R. H. Hardin, Nov 27 2014

Status

approved