A086113 -id:A086113 - cp's OEIS frontend

A207068 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 0 horizontally and 0 0 1 and 1 0 1 vertically.

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 14, 81, 102, 64, 10, 21, 196, 288, 216, 100, 12, 31, 441, 896, 720, 390, 144, 14, 46, 961, 2499, 2688, 1485, 636, 196, 16, 68, 2116, 6634, 8799, 6398, 2709, 966, 256, 18, 100, 4624, 17848, 27063, 23856, 13132, 4536, 1392, 324, 20, 147

Offset: 1

R. H. Hardin Feb 14 2012

Table starts
..2...4....6....9....14.....21.....31......46.......68......100.......147
..4..16...36...81...196....441....961....2116.....4624....10000.....21609
..6..36..102..288...896...2499...6634...17848....47192...122200....315315
..8..64..216..720..2688...8799..27063...84502...257584...762900...2246895
.10.100..390.1485..6398..23856..82739..291364...997288..3297100..10818024
.12.144..636.2709.13132..54684.210118..818892..3093184.11234900..40417797
.14.196..966.4536.24304.111426.468348.1992996..8204200.32364700.126207438
.16.256.1392.7128.41664.208026.947298.4356936.19360144.82226100.344542569

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

R. H. Hardin, Table of n, a(n) for n = 1..1404

Column 2 is A016742
Column 3 is A086113
Row 1 is A038718(n+2)

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 2*n^3 + 6*n^2 - 2*n
k=4: a(n) = (3/4)*n^4 + (15/2)*n^3 + (15/4)*n^2 - 3*n
k=5: a(n) = (7/30)*n^5 + 7*n^4 + (21/2)*n^3 - (56/15)*n
k=6: a(n) = (7/120)*n^6 + (147/40)*n^5 + (49/3)*n^4 + (91/8)*n^3 - (707/120)*n^2 - (91/20)*n
k=7: a(n) = (31/2520)*n^7 + (62/45)*n^6 + (4867/360)*n^5 + (2015/72)*n^4 + (1271/180)*n^3 - (4991/360)*n^2 - (713/140)*n

A207254 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 1 horizontally and 0 1 0 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 8, 36, 36, 10, 10, 64, 102, 100, 16, 12, 100, 216, 370, 256, 26, 14, 144, 390, 940, 1232, 676, 42, 16, 196, 636, 1950, 3776, 4238, 1764, 68, 18, 256, 966, 3560, 9072, 15652, 14406, 4624, 110, 20, 324, 1392, 5950, 18688, 43498, 64176, 49164

Offset: 1

R. H. Hardin Feb 16 2012

Table starts
..2....4.....6......8.....10......12......14.......16.......18.......20
..4...16....36.....64....100.....144.....196......256......324......400
..6...36...102....216....390.....636.....966.....1392.....1926.....2580
.10..100...370....940...1950....3560....5950.....9320....13890....19900
.16..256..1232...3776...9072...18688...34608....59264....95568...146944
.26..676..4238..15652..43498..101036..207298...388232...677898..1119716
.42.1764.14406..64176.206514..541380.1231650..2524704..4777290..8483748
.68.4624.49164.263976.982940.2906592.7328836.16436824.33693660.64313720

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

R. H. Hardin, Table of n, a(n) for n = 1..1057

Column 1 is A006355(n+2)
Column 2 is A206981
Row 2 is A016742
Row 3 is A086113

Empirical for row n:
n=1: a(k) = 2*k
n=2: a(k) = 4*k^2
n=3: a(k) = 2*k^3 + 6*k^2 - 2*k
n=4: a(k) = (5/6)*k^4 + (35/3)*k^3 - (5/6)*k^2 - (5/3)*k
n=5: a(k) = (4/15)*k^5 + (32/3)*k^4 + (44/3)*k^3 - (32/3)*k^2 + (16/15)*k
n=6: a(k) = (13/180)*k^6 + (143/20)*k^5 + (1235/36)*k^4 - (39/4)*k^3 - (377/45)*k^2 + (13/5)*k
n=7: a(k) = (1/60)*k^7 + (77/20)*k^6 + (2527/60)*k^5 + (119/4)*k^4 - (644/15)*k^3 + (42/5)*k^2 + (4/5)*k

A207403 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 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, 12, 81, 102, 64, 10, 16, 144, 289, 216, 100, 12, 20, 256, 612, 729, 390, 144, 14, 25, 400, 1296, 1782, 1521, 636, 196, 16, 30, 625, 2340, 4356, 4212, 2809, 966, 256, 18, 36, 900, 4225, 8910, 11664, 8692, 4761, 1392, 324, 20, 42, 1296

Offset: 1

R. H. Hardin Feb 17 2012

Table starts
..2...4....6....9....12.....16.....20.....25......30......36.......42.......49
..4..16...36...81...144....256....400....625.....900....1296.....1764.....2401
..6..36..102..289...612...1296...2340...4225....6890...11236....17066....25921
..8..64..216..729..1782...4356...8910..18225...33210...60516...101598...170569
.10.100..390.1521..4212..11664..26676..61009..123006..248004...456666...840889
.12.144..636.2809..8692..26896..68060.172225..380970..842724..1690038..3389281
.14.196..966.4761.16284..55696.154580.429025.1033590.2490084..5404650.11730625
.16.256.1392.7569.28362.106276.321110.970225.2529480.6594624.15405432.35988001

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

R. H. Hardin, Table of n, a(n) for n = 1..1512

Column 2 is A016742
Column 3 is A086113
Row 1 is A002620(n+2)
Row 2 is A030179(n+2)
Row 3 is A207118

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 2*n^3 + 6*n^2 - 2*n
k=4: a(n) = n^4 + 6*n^3 + 7*n^2 - 6*n + 1
k=5: a(n) = (1/3)*n^5 + 3*n^4 + (28/3)*n^3 + 7*n^2 - (29/3)*n + 2
k=6: a(n) = (1/9)*n^6 + (4/3)*n^5 + (58/9)*n^4 + (40/3)*n^3 + (49/9)*n^2 - (44/3)*n + 4
k=7: a(n) = (1/36)*n^7 + (4/9)*n^6 + (53/18)*n^5 + (91/9)*n^4 + (589/36)*n^3 + (31/9)*n^2 - (58/3)*n + 6

A207703 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 0 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, 15, 81, 102, 64, 10, 25, 225, 289, 216, 100, 12, 40, 625, 1071, 729, 390, 144, 14, 64, 1600, 3969, 3321, 1521, 636, 196, 16, 104, 4096, 13230, 15129, 8151, 2809, 966, 256, 18, 169, 10816, 44100, 61254, 43681, 17225, 4761, 1392

Offset: 1

R. H. Hardin, Feb 19 2012

Table starts
..2...4....6....9....15.....25......40.......64.......104.......169........273
..4..16...36...81...225....625....1600.....4096.....10816.....28561......74529
..6..36..102..289..1071...3969...13230....44100....153090....531441....1815939
..8..64..216..729..3321..15129...61254...248004...1050282...4447881...18510693
.10.100..390.1521..8151..43681..206910...980100...4863870..24137569..117661437
.12.144..636.2809.17225.105625..571350..3090564..17518470..99301225..552967815
.14.196..966.4761.32775.225625.1369900..8317456..52895444.336392281.2102483853
.16.256.1392.7569.57681.439569.2956980.19891600.140048460.986022801.6826106385

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

R. H. Hardin, Table of n, a(n) for n = 1..650

Column 2 is A016742.
Column 3 is A086113.
Column 4 is A207399.
Row 1 is A006498(n+2).
Row 2 is A189145(n+2).

A208287 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 8, 16, 100, 102, 64, 10, 26, 256, 378, 216, 100, 12, 42, 676, 1260, 984, 390, 144, 14, 68, 1764, 4374, 3984, 2090, 636, 196, 16, 110, 4624, 14946, 16872, 9900, 3900, 966, 256, 18, 178, 12100, 51384, 70216, 49130, 21096, 6650, 1392

Offset: 1

R. H. Hardin Feb 25 2012

Table starts
..2...4....6....10....16.....26......42.......68.......110........178
..4..16...36...100...256....676....1764.....4624.....12100......31684
..6..36..102...378..1260...4374...14946....51384....176238.....605022
..8..64..216...984..3984..16872...70216...294192...1229400....5142728
.10.100..390..2090..9900..49130..239490..1175440...5754050...28195750
.12.144..636..3900.21096.119580..665892..3733080..20874900..116842500
.14.196..966..6650.40376.256774.1604862.10095932..63357434..397965218
.16.256.1392.10608.71360.502416.3478160.24229696.168399632.1171405168

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

R. H. Hardin, Table of n, a(n) for n = 1..1106

Column 2 is A016742
Column 3 is A086113
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A060521

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 2*n^3 + 6*n^2 - 2*n
k=4: a(n) = (4/3)*n^4 + 10*n^3 + (2/3)*n^2 - 2*n
k=5: a(n) = (5/6)*n^5 + 9*n^4 + (91/6)*n^3 - 9*n^2
k=6: a(n) = (8/15)*n^6 + (23/3)*n^5 + (82/3)*n^4 + (1/3)*n^3 - (178/15)*n^2 + 2*n
k=7: a(n) = (61/180)*n^7 + (121/20)*n^6 + (1157/36)*n^5 + (425/12)*n^4 - (2923/90)*n^3 - (22/15)*n^2 + 2*n

A209650 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 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, 14, 81, 102, 64, 10, 22, 196, 270, 216, 100, 12, 35, 484, 798, 630, 390, 144, 14, 56, 1225, 2354, 2156, 1215, 636, 196, 16, 90, 3136, 7210, 7128, 4690, 2079, 966, 256, 18, 145, 8100, 22232, 24990, 16830, 8904, 3276, 1392, 324, 20, 234

Offset: 1

R. H. Hardin Mar 11 2012

Table starts
..2...4....6....9....14.....22.....35......56.......90......145.......234
..4..16...36...81...196....484...1225....3136.....8100....21025.....54756
..6..36..102..270...798...2354...7210...22232....69570...218950....693810
..8..64..216..630..2156...7128..24990...87136...311040..1112150...4018716
.10.100..390.1215..4690..16830..65765..251160...994050..3911375..15639390
.12.144..636.2079..8904..34012.145775..597856..2579940.10954895..47622744
.14.196..966.3276.15386..61754.287140.1247736..5805450.26247900.122620446
.16.256.1392.4860.24808.103664.518700.2364992.11769120.56106300.279344520

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

R. H. Hardin, Table of n, a(n) for n = 1..2828

Column 2 is A016742
Column 3 is A086113
Row 1 is A001611(n+2)
Row 2 is A207436
Row 3 is A207747

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 2*n^3 + 6*n^2 - 2*n
k=4: a(n) = 9*n^3 + (9/2)*n^2 - (9/2)*n
k=5: a(n) = (7/2)*n^4 + 21*n^3 - (7/2)*n^2 - 7*n
k=6: a(n) = 22*n^4 + (88/3)*n^3 - 22*n^2 - (22/3)*n
k=7: a(n) = 7*n^5 + 70*n^4 + (35/3)*n^3 - (105/2)*n^2 - (7/6)*n

A032260 Number of n X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.

Original entry on oeis.org

2, 16, 102, 528, 2470, 11016, 47950, 205792, 874998, 3694920, 15519262, 64899456, 270415262, 1123264408, 4653525150, 19234571968, 79342610902, 326704870152, 1343120023678, 5513861152000, 22606830725598, 92580354402712, 378737813468542, 1547884976787648

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jun 25 2003

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Don Coppersmith, Ponder This: IBM Research Monthly Puzzles, March 2004 challenge

The number of n X n 0, 1 matrices such that each row and each column is increasing is in sequence A000984.

Cf. A000984, A062528, A045992, A016742, A086113 - A086115.

a(n) = 2*n*(binomial(2*n, n)-n). G.f.: 4*x/(1-4*x)^(3/2)-2*x*(1+x)/(1-x)^3. - Vladimir Baltic and Vladeta Jovovic, Jul 10 2003

Extended by Vladimir Baltic and Vladeta Jovovic, Jul 10 2003

More terms from Eric M. Schmidt, May 01 2013

A202052 T(n,k)=Number of (n+2)X(k+2) binary arrays avoiding patterns 001 and 110 in rows and columns.

Original entry on oeis.org

102, 216, 216, 390, 528, 390, 636, 1080, 1080, 636, 966, 1968, 2470, 1968, 966, 1392, 3304, 4980, 4980, 3304, 1392, 1926, 5216, 9170, 11016, 9170, 5216, 1926, 2580, 7848, 15760, 22092, 22092, 15760, 7848, 2580, 3366, 11360, 25650, 41088, 47950, 41088

Offset: 1

R. H. Hardin Dec 10 2011

Table starts
..102...216...390....636....966....1392....1926.....2580.....3366.....4296
..216...528..1080...1968...3304....5216....7848....11360....15928....21744
..390..1080..2470...4980...9170...15760...25650....39940....59950....87240
..636..1968..4980..11016..22092...41088...71964...120000...192060...296880
..966..3304..9170..22092..47950...95984..180054...320180...544390...890904
.1392..5216.15760..41088..95984..205792..411696...777760..1400080..2418432
.1926..7848.25650..71964.180054..411696..874998..1750140..3325410..6046344
.2580.11360.39940.120000.320180..777760.1750140..3694920..7390020.14108400
.3366.15928.59950.192060.544390.1400080.3325410..7390020.15519262.31038744
.4296.21744.87240.296880.890904.2418432.6046344.14108400.31038744.64899456

			Some solutions for n=5 k=3
..0..0..0..0..0....1..0..1..1..1....0..1..0..1..1....0..1..0..1..1
..1..1..1..1..1....0..1..0..0..0....1..0..1..0..0....1..0..0..0..0
..0..0..0..0..0....1..0..1..1..1....0..1..0..1..0....0..1..0..1..1
..0..1..1..1..1....0..0..0..0..0....0..0..0..0..0....1..0..0..0..0
..0..1..0..0..0....1..0..1..1..1....0..1..0..1..0....0..1..0..1..1
..0..1..0..1..1....0..0..0..0..0....0..1..0..1..0....0..0..0..0..0
..0..1..0..1..0....1..0..1..0..0....0..1..0..1..0....0..1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..511

Column 1 is A086113(n+2)
Column 2 is A086114(n+2)
Column 3 is A086115(n+2)
Diagonal is A032260(n+2)

Empirical (via A086113): T(n,k)=2*(n+2)*(2*binomial(n+k+3,n+2)-k-2)
Empirical for columns:
T(n,1) = 2*n^3 + 18*n^2 + 46*n + 36
T(n,2) = (2/3)*n^4 + (28/3)*n^3 + (142/3)*n^2 + (284/3)*n + 64
T(n,3) = (1/6)*n^5 + (10/3)*n^4 + (155/6)*n^3 + (290/3)*n^2 + 164*n + 100
T(n,4) = (1/30)*n^6 + (9/10)*n^5 + (59/6)*n^4 + (111/2)*n^3 + (2552/15)*n^2 + (1278/5)*n + 144
T(n,5) = (1/180)*n^7 + (7/36)*n^6 + (511/180)*n^5 + (805/36)*n^4 + (4606/45)*n^3 + (2443/9)*n^2 + (1854/5)*n + 196
T(n,6) = (1/1260)*n^8 + (11/315)*n^7 + (59/90)*n^6 + (308/45)*n^5 + (7807/180)*n^4 + (7667/45)*n^3 + (14139/35)*n^2 + (17876/35)*n + 256
T(n,7) = (1/10080)*n^9 + (3/560)*n^8 + (211/1680)*n^7 + (67/40)*n^6 + (6709/480)*n^5 + (6041/80)*n^4 + (663941/2520)*n^3 + (79913/140)*n^2 + (4735/7)*n + 324

A086115 Number of 5 X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.

Original entry on oeis.org

10, 100, 390, 1080, 2470, 4980, 9170, 15760, 25650, 39940, 59950, 87240, 123630, 171220, 232410, 309920, 406810, 526500, 672790, 849880, 1062390, 1315380, 1614370, 1965360, 2374850, 2849860, 3397950, 4027240, 4746430, 5564820

Offset: 1

Vladimir Baltic, Vladeta Jovovic, Jul 10 2003

Don Coppersmith, Ponder This: IBM Research Monthly Puzzles, March 2004 challenge
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A032260, A016742, A086113, A086114.

a(n) = (1/6)*n*(n^4+10*n^3+35*n^2+50*n-36). More generally, number of m X n (0, 1) matrices such that each row and each column is increasing or decreasing is 2*n*(2*binomial(n+m-1, n)-m) = 4/Beta(m, n)-2*m*n.

G.f.: -10*x*(x^4-4*x^3+6*x^2-4*x-1) / (x-1)^6. [Colin Barker, Feb 22 2013]

A086114 Number of 4 X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.

Original entry on oeis.org

8, 64, 216, 528, 1080, 1968, 3304, 5216, 7848, 11360, 15928, 21744, 29016, 37968, 48840, 61888, 77384, 95616, 116888, 141520, 169848, 202224, 239016, 280608, 327400, 379808, 438264, 503216, 575128, 654480, 741768, 837504, 942216, 1056448

Offset: 1

Vladimir Baltic, Vladeta Jovovic, Jul 10 2003

Don Coppersmith, Ponder This: IBM Research Monthly Puzzles, March 2004 challenge
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A032260, A016742, A086113, A086115.

a(n) = (2/3)*n*(n^3+6*n^2+11*n-6). More generally, number of m X n (0, 1) matrices such that each row and each column is increasing or decreasing is 2*n*(2*binomial(n+m-1, n)-m) = 4/Beta(m, n)-2*m*n.

G.f.: -8*x*(x^3-3*x^2+3*x+1) / (x-1)^5. [Colin Barker, Feb 22 2013]

cp's OEIS Frontend

A207068 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 0 horizontally and 0 0 1 and 1 0 1 vertically.

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A207254 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 1 horizontally and 0 1 0 and 1 0 1 vertically.

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A207403 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 0 1 and 1 1 0 vertically.

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A207703 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 1 0 vertically.

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A208287 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 1 1 0 vertically.

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A209650 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 0 1 and 1 0 1 vertically.

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A032260 Number of n X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.

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A202052 T(n,k)=Number of (n+2)X(k+2) binary arrays avoiding patterns 001 and 110 in rows and columns.

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A086115 Number of 5 X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.

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A086114 Number of 4 X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.

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