A171861 -id:A171861 - cp's OEIS frontend

A191670 Dispersion of A042968 (>1 and congruent to 1 or 2 or 3 mod 4), by antidiagonals.

1, 2, 4, 3, 6, 8, 5, 9, 11, 12, 7, 13, 15, 17, 16, 10, 18, 21, 23, 22, 20, 14, 25, 29, 31, 30, 27, 24, 19, 34, 39, 42, 41, 37, 33, 28, 26, 46, 53, 57, 55, 50, 45, 38, 32, 35, 62, 71, 77, 74, 67, 61, 51, 43, 36, 47, 83, 95, 103, 99, 90, 82, 69, 58, 49, 40, 63

Offset: 1

Clark Kimberling, Jun 11 2011

For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
...
A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
   a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
   a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

			Northwest corner:
1....2....3....5....7
4....6....9....13...18
8....11...15...21...29
12...17...23...31...42
16...22...30...41...55

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Row 1: A155167, Row 2: A171861.

Cf. A008486, A042968, A191673, A191452.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a = 2; b = 3; c2 = 5; m[n_] := If[Mod[n, 3] == 0, 1, 0];
f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
Table[f[n], {n, 1, 30}] (* A042968 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191670 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191670 *)

A207024 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 13, 81, 90, 64, 10, 18, 169, 252, 168, 100, 12, 25, 324, 624, 558, 270, 144, 14, 34, 625, 1350, 1586, 1035, 396, 196, 16, 46, 1156, 3025, 3726, 3315, 1719, 546, 256, 18, 62, 2116, 6256, 9450, 8280, 6123, 2646, 720, 324, 20, 83, 3844

Offset: 1

R. H. Hardin, Feb 14 2012

Table starts
..2...4...6....9....13....18.....25.....34......46......62.......83......111
..4..16..36...81...169...324....625...1156....2116....3844.....6889....12321
..6..36..90..252...624..1350...3025...6256...12788...25792....50630....99012
..8..64.168..558..1586..3726...9450..21318...47518..104470...220531...464202
.10.100.270.1035..3315..8280..23400..56814..136114..322834...725005..1627260
.12.144.396.1719..6123.16038..49925.129302..329498..836938..1984447..4716834
.14.196.546.2646.10374.28224..95900.263228..707756.1914436..4765030.11929281
.16.256.720.3852.16484.46260.170300.492932.1389476.3984244.10362550.27202659

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

R. H. Hardin, Table of n, a(n) for n = 1..1984

Column 2 is A016742.
Column 3 is A152746.
Row 1 is A171861(n+1).

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) = 6*n^3 + (27/2)*n^2 - (21/2)*n
k=5: a(n) = (13/6)*n^4 + 13*n^3 + (52/3)*n^2 - (39/2)*n
k=6: a(n) = (33/4)*n^4 + (45/2)*n^3 + (75/4)*n^2 - (63/2)*n
k=7: a(n) = (55/24)*n^5 + (75/4)*n^4 + (275/8)*n^3 + (75/4)*n^2 - (295/6)*n

A207111 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 13, 81, 98, 64, 10, 18, 169, 271, 200, 100, 12, 25, 324, 677, 643, 350, 144, 14, 34, 625, 1504, 1835, 1271, 556, 196, 16, 46, 1156, 3399, 4534, 4047, 2239, 826, 256, 18, 62, 2116, 7220, 11511, 10898, 7837, 3641, 1168, 324, 20, 83, 3844

Offset: 1

R. H. Hardin Feb 15 2012

Table starts
..2...4....6....9....13....18.....25.....34......46......62.......83......111
..4..16...36...81...169...324....625...1156....2116....3844.....6889....12321
..6..36...98..271...677..1504...3399...7220...15184...31664....64749...132543
..8..64..200..643..1835..4534..11511..27012...62814..144676...325111...733469
.10.100..350.1271..4047.10898..30415..77326..194952..486102..1177409..2870021
.12.144..556.2239..7837.22714..68737.187054..505040.1346150..3472283..9030485
.14.196..826.3641.13863.42874.139341.402498.1153962.3259098..8878431.24420005
.16.256.1168.5581.22931.75198.260597.794118.2402578.7142988.20426983.59031673

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

R. H. Hardin, Table of n, a(n) for n = 1..612

Column 2 is A016742
Row 1 is A171861(n+1)
Row 2 is A207025

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = (4/3)*n^3 + 8*n^2 - (10/3)*n
k=4: a(n) = (5/12)*n^4 + (13/2)*n^3 + (115/12)*n^2 - (17/2)*n + 1
k=5: a(n) = (7/60)*n^5 + (8/3)*n^4 + (185/12)*n^3 + (19/3)*n^2 - (218/15)*n + 3
k=6: a(n) = (7/360)*n^6 + (77/120)*n^5 + (635/72)*n^4 + (623/24)*n^3 - (511/180)*n^2 - (103/5)*n + 6
k=7: a(n) = (1/280)*n^7 + (7/45)*n^6 + (47/15)*n^5 + (206/9)*n^4 + (4111/120)*n^3 - (1037/45)*n^2 - (4493/210)*n + 9

A207242 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 0 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 102, 81, 14, 18, 169, 283, 287, 196, 21, 25, 324, 699, 987, 882, 441, 31, 34, 625, 1526, 2884, 3866, 2491, 961, 46, 46, 1156, 3355, 7165, 13876, 13494, 6759, 2116, 68, 62, 2116, 6888, 17929, 40838, 58026, 44730, 18528

Offset: 1

R. H. Hardin Feb 16 2012

Table starts
..2....4.....6......9.....13......18.......25.......34........46........62
..4...16....36.....81....169.....324......625.....1156......2116......3844
..6...36...102....283....699....1526.....3355.....6888.....13954.....27816
..9...81...287....987...2884....7165....17929....40646.....90602....196548
.14..196...882...3866..13876...40838...122197...323039....839664...2118081
.21..441..2491..13494..58026..197256...683257..2037721...5952936..16732138
.31..961..6759..44730.228110..886959..3510185.11662500..37779767.116530737
.46.2116.18528.150608.919413.4132837.18982182.71594896.262849612.914523558

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

R. H. Hardin, Table of n, a(n) for n = 1..449

Column 1 is A038718(n+2)
Column 2 is A207069
Row 1 is A171861(n+1)
Row 2 is A207025

A207269 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 82, 81, 14, 18, 169, 217, 193, 196, 21, 25, 324, 499, 611, 488, 441, 31, 34, 625, 1014, 1602, 1910, 1087, 961, 46, 46, 1156, 2141, 3513, 5904, 5132, 2305, 2116, 68, 62, 2116, 4188, 8327, 14184, 18055, 13067, 4932, 4624, 100, 83

Offset: 1

R. H. Hardin Feb 16 2012

Table starts
..2....4....6.....9.....13.....18......25......34.......46.......62.......83
..4...16...36....81....169....324.....625....1156.....2116.....3844.....6889
..6...36...82...217....499...1014....2141....4188.....8150....15670....29517
..9...81..193...611...1602...3513....8327...17568....36988....76723...153865
.14..196..488..1910...5904..14184...38911...90670...211626...485438..1060958
.21..441.1087..5132..18055..45520..140295..349415...877101..2164569..5025447
.31..961.2305.13067..52240.135686..467686.1233127..3284096..8655283.21076355
.46.2116.4932.33937.156473.417772.1623957.4556171.12881467.36421182.93059536

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

R. H. Hardin, Table of n, a(n) for n = 1..881

Column 1 is A038718(n+2)
Column 2 is A207069
Row 1 is A171861(n+1)
Row 2 is A207025

A207346 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 0 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 10, 13, 81, 84, 100, 16, 18, 169, 192, 292, 256, 26, 25, 324, 426, 828, 912, 676, 42, 34, 625, 858, 2190, 3130, 2812, 1764, 68, 46, 1156, 1704, 5290, 9668, 11230, 8928, 4624, 110, 62, 2116, 3330, 12292, 27022, 41112, 43260, 28152

Offset: 1

R. H. Hardin Feb 17 2012

Table starts
..2....4.....6......9.....13......18.......25.......34........46........62
..4...16....36.....81....169.....324......625.....1156......2116......3844
..6...36....84....192....426.....858.....1704.....3330......6390.....12150
.10..100...292....828...2190....5290....12292....27978.....62574....136978
.16..256...912...3130...9668...27022....71344...185624....468864...1161400
.26..676..2812..11230..41112..128292...388134..1134282...3225374...9002616
.42.1764..8928..43260.186002..684324..2367656..8000806..26057006..82907430
.68.4624.28152.163710.828530.3524736.13950908.53882188.199473062.720751822

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

R. H. Hardin, Table of n, a(n) for n = 1..391

Column 1 is A006355(n+2)
Column 2 is A206981
Row 1 is A171861(n+1)
Row 2 is A207025

A207589 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 60, 81, 13, 26, 256, 144, 144, 169, 18, 42, 676, 324, 432, 312, 324, 25, 68, 1764, 756, 1098, 1014, 612, 625, 34, 110, 4624, 1728, 3150, 3094, 2232, 1250, 1156, 46, 178, 12100, 3996, 8244, 9698, 7272, 5000, 2516, 2116, 62

Offset: 1

R. H. Hardin, Feb 19 2012

Table starts
..2....4....6....10....16.....26.....42......68......110......178.......288
..4...16...36...100...256....676...1764....4624....12100....31684.....82944
..6...36...60...144...324....756...1728....3996.....9180....21168.....48708
..9...81..144...432..1098...3150...8244...23202....61560...171468....458640
.13..169..312..1014..3094...9698..30056...93782...291304...908102...2822456
.18..324..612..2232..7272..25776..85536..300096..1004364..3501756..11782620
.25..625.1250..5000.18150..70800.263550.1018700..3824100.14714500..55475900
.34.1156.2516.11220.46716.204476.876860.3832616.16578128.72435844.314466680

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

R. H. Hardin, Table of n, a(n) for n = 1..971

Column 1 is A171861(n+1).
Column 2 is A207025.
Row 1 is A006355(n+2).
Row 2 is A206981.

A207368 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 90, 81, 12, 18, 169, 252, 225, 144, 16, 25, 324, 624, 784, 420, 256, 20, 34, 625, 1350, 2304, 1736, 784, 400, 25, 46, 1156, 3025, 5625, 5856, 3844, 1260, 625, 30, 62, 2116, 6256, 14641, 15525, 14884, 7130, 2025, 900, 36, 83

Offset: 1

R. H. Hardin Feb 17 2012

Table starts
..2...4....6.....9....13.....18.....25......34......46.......62.......83
..4..16...36....81...169....324....625....1156....2116.....3844.....6889
..6..36...90...252...624...1350...3025....6256...12788....25792....50630
..9..81..225...784..2304...5625..14641...33856...77284...173056...372100
.12.144..420..1736..5856..15525..45738..115368..287174...700960..1620770
.16.256..784..3844.14884..42849.142884..393129.1067089..2839225..7059649
.20.400.1260..7130.31110..95220.353808.1047717.3056647..8773795.23208895
.25.625.2025.13225.65025.211600.876096.2792241.8755681.27112849.76300225

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

R. H. Hardin, Table of n, a(n) for n = 1..684

Column 1 is A002620(n+2)
Column 2 is A030179(n+2)
Row 1 is A171861(n+1)
Row 2 is A207025
Row 3 is A207026

A207467 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 1 0 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 10, 13, 81, 98, 100, 16, 18, 169, 271, 358, 256, 26, 25, 324, 677, 1307, 1152, 676, 42, 34, 625, 1504, 4219, 5369, 3910, 1764, 68, 46, 1156, 3399, 11760, 21517, 23645, 12994, 4624, 110, 62, 2116, 7220, 33393, 71928, 119965, 101233

Offset: 1

R. H. Hardin Feb 18 2012

Table starts
..2....4.....6......9......13.......18........25........34.........46
..4...16....36.....81.....169......324.......625......1156.......2116
..6...36....98....271.....677.....1504......3399......7220......15184
.10..100...358...1307....4219....11760.....33393.....88198.....229458
.16..256..1152...5369...21517....71928....247631....778010....2406006
.26..676..3910..23645..119965...491948...2090927...7990970...29983194
.42.1764.12994.101233..644401..3205650..16675001..76629370..345252578
.68.4624.43596.439063.3523073.21396734.136803627.760840082.4139073950

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

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 1 is A006355(n+2)
Column 2 is A206981
Row 1 is A171861(n+1)
Row 2 is A207025
Row 3 is A207112

A207488 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 1 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 82, 81, 14, 18, 169, 217, 177, 196, 22, 25, 324, 499, 530, 408, 484, 35, 34, 625, 1014, 1322, 1459, 942, 1225, 56, 46, 1156, 2141, 2749, 4041, 3947, 2233, 3136, 90, 62, 2116, 4188, 6217, 8808, 12151, 11306, 5348, 8100, 145, 83

Offset: 1

R. H. Hardin Feb 18 2012

Table starts
..2....4....6.....9.....13.....18.....25......34......46.......62.......83
..4...16...36....81....169....324....625....1156....2116.....3844.....6889
..6...36...82...217....499...1014...2141....4188....8150....15670....29517
..9...81..177...530...1322...2749...6217...12712...25908....52474...103704
.14..196..408..1459...4041...8808..21991...48044..105558...230156...490287
.22..484..942..3947..12151..26980..73325..170276..397692...935992..2150515
.35.1225.2233.11306..39050..89417.268275..667760.1691526..4324858.10910668
.56.3136.5348.32445.126411.292136.959855.2555724.6929562.19404152.53733153

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

R. H. Hardin, Table of n, a(n) for n = 1..577

Column 1 is A001611(n+2)
Column 2 is A207436
Row 1 is A171861(n+1)
Row 2 is A207025
Row 3 is A207270

cp's OEIS Frontend

A191670 Dispersion of A042968 (>1 and congruent to 1 or 2 or 3 mod 4), by antidiagonals.

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A207024 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

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A207111 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 1 0 vertically.

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A207242 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 0 1 0 vertically.

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A207269 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 1 1 and 1 0 1 vertically.

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A207346 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 0 and 1 1 1 vertically.

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A207589 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 1 1 1 vertically.

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A207368 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A207467 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 1 0 and 1 0 1 vertically.

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A207488 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 1 1 and 1 1 0 vertically.

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