A203094 -id:A203094 - cp's OEIS frontend

A231839 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal and antidiagonal neighbors.

4, 4, 16, 16, 50, 64, 50, 188, 422, 256, 144, 760, 4508, 3823, 1024, 422, 3309, 52411, 111621, 34350, 4096, 1268, 14666, 678660, 3477361, 2836554, 308419, 16384, 3823, 64607, 8887871, 124132900, 241961326, 71178861, 2771101, 65536, 11472, 283479

Offset: 1

R. H. Hardin, Nov 14 2013

Table starts
.......4..........4.............16.................50....................144
......16.........50............188................760...................3309
......64........422...........4508..............52411.................678660
.....256.......3823.........111621............3477361..............124132900
....1024......34350........2836554..........241961326............24188209253
....4096.....308419.......71178861........16599585680..........4666623161419
...16384....2771101.....1792092360......1140658285204........899426070636904
...65536...24892609....45099279326.....78428361897720.....173546274977761257
..262144..223618304..1134900171250...5390322528656652...33474310504831841795
.1048576.2008825312.28560684486812.370517687958114665.6456965889651937136227

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

R. H. Hardin, Table of n, a(n) for n = 1..178

Column 1 is A000302

Row 1 is A203094 for n>1

Empirical for column k:
k=1: a(n) = 4*a(n-1)
k=2: a(n) = 4*a(n-1) +34*a(n-2) +86*a(n-3) +91*a(n-4) +46*a(n-5) +11*a(n-6) +a(n-7)
k=3: [order 10] for n>11
k=4: [order 29] for n>30
k=5: [order 82] for n>83
Empirical for row n:
n=1: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7) for n>8
n=2: [order 31] for n>32

A217954 T(n,k) = number of n-element 0..3 arrays with each element the minimum of k adjacent elements of a random 0..3 array of n+k-1 elements.

Original entry on oeis.org

4, 4, 16, 4, 16, 64, 4, 16, 50, 256, 4, 16, 50, 144, 1024, 4, 16, 50, 130, 422, 4096, 4, 16, 50, 130, 310, 1268, 16384, 4, 16, 50, 130, 296, 736, 3823, 65536, 4, 16, 50, 130, 296, 624, 1821, 11472, 262144, 4, 16, 50, 130, 296, 610, 1289, 4673, 34350, 1048576, 4, 16

Offset: 1

R. H. Hardin, suggestion that the diagonal might be a polynomial from L. Edson Jeffery in the Sequence Fans Mailing List, Oct 15 2012

See A228461 for comments on the definition. - N. J. A. Sloane, Sep 02 2013
Table starts
........4......4......4.....4.....4.....4.....4.....4.....4.....4.....4.....4
.......16.....16.....16....16....16....16....16....16....16....16....16....16
.......64.....50.....50....50....50....50....50....50....50....50....50....50
......256....144....130...130...130...130...130...130...130...130...130...130
.....1024....422....310...296...296...296...296...296...296...296...296...296
.....4096...1268....736...624...610...610...610...610...610...610...610...610
....16384...3823...1821..1289..1177..1163..1163..1163..1163..1163..1163..1163
....65536..11472...4673..2741..2209..2097..2083..2083..2083..2083..2083..2083
...262144..34350..12107..6134..4202..3670..3558..3544..3544..3544..3544..3544
..1048576.102896..31103.14269..8366..6434..5902..5790..5776..5776..5776..5776
..4194304.308419..79039.33577.17569.11666..9734..9202..9090..9076..9076..9076
.16777216.924532.199819.78304.38251.22313.16410.14478.13946.13834.13820.13820

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

R. H. Hardin, Table of n, a(n) for n = 1..1275

Column 2 is A203094(n+1). A217949 is also a column. Cf. A228461, A217883.

Empirical for column k:
k=2: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7)
k=3: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) +5*a(n-4) -4*a(n-5) +6*a(n-6) +4*a(n-7) +2*a(n-9) +a(n-10)
k=4: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-5) -4*a(n-6) +6*a(n-7) +4*a(n-8) +5*a(n-9) +a(n-10) +3*a(n-11) +2*a(n-12) +a(n-13)
k=5: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-6) -4*a(n-7) +6*a(n-8) +4*a(n-9) +5*a(n-10) +6*a(n-11) +2*a(n-12) +4*a(n-13) +3*a(n-14) +2*a(n-15) +a(n-16)
k=6: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-7) -4*a(n-8) +6*a(n-9) +4*a(n-10) +5*a(n-11) +6*a(n-12) +7*a(n-13) +3*a(n-14) +5*a(n-15) +4*a(n-16) +3*a(n-17) +2*a(n-18) +a(n-19)
k=7: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-8) -4*a(n-9) +6*a(n-10) +4*a(n-11) +5*a(n-12) +6*a(n-13) +7*a(n-14) +8*a(n-15) +4*a(n-16) +6*a(n-17) +5*a(n-18) +4*a(n-19) +3*a(n-20) +2*a(n-21) +a(n-22)
Diagonal: a(n) = (1/720)*n^6 + (1/48)*n^5 + (23/144)*n^4 + (9/16)*n^3 + (241/180)*n^2 + (11/12)*n + 1

A231940 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal, diagonal and antidiagonal neighbors.

Original entry on oeis.org

4, 4, 16, 16, 84, 64, 50, 668, 318, 256, 144, 5070, 8426, 1328, 1024, 422, 42104, 206808, 152180, 6064, 4096, 1268, 326010, 4736026, 11159202, 2462572, 26918, 16384, 3823, 2511252, 94464137, 691418144, 518972238, 36885538, 116909, 65536, 11472

Offset: 1

R. H. Hardin, Nov 15 2013

Table starts
.......4.......4............16...............50................144
......16......84...........668.............5070..............42104
......64.....318..........8426...........206808............4736026
.....256....1328........152180.........11159202..........691418144
....1024....6064.......2462572........518972238........86074040354
....4096...26918......36885538......23280281589.....10417626293694
...16384..116909.....586971925....1098832065447...1320620287047433
...65536..511264....9394148948...52087504055935.167980047970274816
..262144.2248196..147360195020.2432351670277323
.1048576.9868600.2323912599668

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

R. H. Hardin, Table of n, a(n) for n = 1..84

Column 1 is A000302
Column 2 is A231741
Row 1 is A203094 for n>1

Empirical for column k:
k=1: a(n) = 4*a(n-1)
k=2: [order 19] for n>20
k=3: [order 65] for n>66
Empirical for row n:
n=1: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7) for n>8
n=2: [order 15]

A231700 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal, vertical and antidiagonal neighbors.

Original entry on oeis.org

4, 4, 4, 16, 28, 16, 50, 272, 272, 50, 144, 1998, 5972, 1998, 144, 422, 13260, 115583, 115583, 13260, 422, 1268, 94996, 2049855, 6074096, 2049855, 94996, 1268, 3823, 691229, 37872601, 286808607, 286808607, 37872601, 691229, 3823, 11472, 4926082

Offset: 1

R. H. Hardin, Nov 12 2013

Table starts
.....4.........4............16................50....................144
.....4........28...........272..............1998..................13260
....16.......272..........5972............115583................2049855
....50......1998........115583...........6074096..............286808607
...144.....13260.......2049855.........286808607............34764511754
...422.....94996......37872601.......14007173906..........4333974317355
..1268....691229.....711067486......696904625544........556835183317668
..3823...4926082...13223846355....34390368202345......71067313709831125
.11472..35082734..245394073602..1693611631652448....9020067185976407959
.34350.251198534.4563802124823.83546478885400765.1146622410485781036080

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

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A203094 for n>1

Empirical for column k:
k=1: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7) for n>8
k=2: [order 22]

A231586 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal, vertical, diagonal and antidiagonal neighbors.

Original entry on oeis.org

4, 4, 4, 16, 28, 16, 50, 124, 124, 50, 144, 602, 2352, 602, 144, 422, 2776, 32650, 32650, 2776, 422, 1268, 12922, 381076, 1245468, 381076, 12922, 1268, 3823, 60720, 5618444, 38768838, 38768838, 5618444, 60720, 3823, 11472, 286047, 82479418, 1453028265

Offset: 1

R. H. Hardin, Nov 11 2013

Table starts
.....4.......4...........16...............50................144
.....4......28..........124..............602...............2776
....16.....124.........2352............32650.............381076
....50.....602........32650..........1245468...........38768838
...144....2776.......381076.........38768838.........2912579313
...422...12922......5618444.......1453028265.......260228939106
..1268...60720.....82479418......57099276134.....24813120747919
..3823..286047...1142396851....2144257250786...2275463989704521
.11472.1335296..16061107556...79701582158514.205616812796559119
.34350.6256326.231334122108.3015862229160377

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

R. H. Hardin, Table of n, a(n) for n = 1..97

Column 1 is A203094 for n>1

Empirical for column k:
k=1: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7) for n>8
k=2: [order 16]
k=3: [order 72] for n>73

A231746 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal and vertical neighbors.

Original entry on oeis.org

4, 4, 4, 16, 84, 16, 50, 318, 318, 50, 144, 1328, 4430, 1328, 144, 422, 6064, 60806, 60806, 6064, 422, 1268, 26918, 784076, 2154900, 784076, 26918, 1268, 3823, 116909, 9945132, 71742015, 71742015, 9945132, 116909, 3823, 11472, 511264, 126926437

Offset: 1

R. H. Hardin, Nov 13 2013

Table starts
....4......4.........16............50..............144..................422
....4.....84........318..........1328.............6064................26918
...16....318.......4430.........60806...........784076..............9945132
...50...1328......60806.......2154900.........71742015...........2426463539
..144...6064.....784076......71742015.......6425495277.........598711367728
..422..26918....9945132....2426463539.....598711367728......155985773478611
.1268.116909..126926437...82701946547...56006640316980....40539474933206048
.3823.511264.1625269595.2814528154294.5220406326242670.10471241201508754882

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

R. H. Hardin, Table of n, a(n) for n = 1..111

Column 1 is A203094 for n>1

Empirical for column k:
k=1: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7) for n>8
k=2: [order 19] for n>20
k=3: [order 87] for n>88

A238287 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with no element greater than all horizontal neighbors or less than all vertical neighbors.

Original entry on oeis.org

4, 16, 16, 50, 204, 50, 144, 1844, 1844, 144, 422, 13948, 42084, 13948, 422, 1268, 105862, 737366, 737366, 105862, 1268, 3823, 850420, 12926271, 27913368, 12926271, 850420, 3823, 11472, 6953993, 245920800, 1058583000, 1058583000, 245920800

Offset: 1

R. H. Hardin, Feb 22 2014

Table starts
......4.........16.............50................144.................422
.....16........204...........1844..............13948..............105862
.....50.......1844..........42084.............737366............12926271
....144......13948.........737366...........27913368..........1058583000
....422.....105862.......12926271.........1058583000.........87389474502
...1268.....850420......245920800........44441926832.......8155057418133
...3823....6953993.....4810332239......1931022884277.....790756782942944
..11472...56279542....92790334588.....82508389421874...75089860262272452
..34350..451637564..1767175335274...3470891253761820.7004416705370946171
.102896.3624058880.33631393265283.145872518046542058

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

R. H. Hardin, Table of n, a(n) for n = 1..97

Column 1 is A203094(n+1)

Empirical for column k:
k=1: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7)
k=2: [order 25]

cp's OEIS Frontend

A231839 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal and antidiagonal neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A217954 T(n,k) = number of n-element 0..3 arrays with each element the minimum of k adjacent elements of a random 0..3 array of n+k-1 elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A231940 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal, diagonal and antidiagonal neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A231700 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal, vertical and antidiagonal neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A231586 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal, vertical, diagonal and antidiagonal neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A231746 T(n,k)=Number of nXk 0..3 arrays with no element less than a strict majority of its horizontal and vertical neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A238287 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with no element greater than all horizontal neighbors or less than all vertical neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula