cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A280673 T(n,k)=Number of nXk 0..2 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

1, 2, 2, 4, 11, 5, 11, 59, 82, 14, 30, 338, 858, 612, 41, 82, 1917, 10205, 12484, 4568, 122, 224, 10893, 119440, 310365, 181640, 34096, 365, 612, 61880, 1401470, 7533245, 9439606, 2642832, 254496, 1094, 1672, 351541, 16438612, 183331502, 474736149
Offset: 1

Views

Author

R. H. Hardin, Jan 07 2017

Keywords

Comments

Table starts
....1.........2............4..............11...............30................82
....2........11...........59.............338.............1917.............10893
....5........82..........858...........10205...........119440...........1401470
...14.......612........12484..........310365..........7533245.........183331502
...41......4568.......181640.........9439606........474736149.......23952262535
..122.....34096......2642832.......287101721......29920114246.....3130289979912
..365....254496.....38452768......8732086113....1885698283255...409089889172506
.1094...1899584....559481408....265582964074..118845116023725.53463025958093933
.3281..14178688...8140361856...8077601392565.7490149091439288
.9842.105831168.118440917248.245677069239189

Examples

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

Links

Crossrefs

Column 1 is A007051(n-1).
Column 2 is A209094.
Row 1 is A021006(n-3).

Formula

Empirical for column k:
k=1: a(n) = 4*a(n-1) -3*a(n-2)
k=2: a(n) = 8*a(n-1) -4*a(n-2) for n>3
k=3: a(n) = 14*a(n-1) +8*a(n-2)
k=4: a(n) = 29*a(n-1) +44*a(n-2) -27*a(n-3) -81*a(n-4) for n>5
k=5: [order 8] for n>9
k=6: [order 20] for n>22
Empirical for row n:
n=1: a(n) = 2*a(n-1) +2*a(n-2) for n>4
n=2: a(n) = 5*a(n-1) +6*a(n-2) -11*a(n-3) -7*a(n-4) +4*a(n-5) for n>6
n=3: [order 18] for n>20
n=4: [order 73] for n>78

A281605 T(n,k)=Number of nXk 0..2 arrays with no element equal to more than one of its horizontal, diagonal or antidiagonal neighbors and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

1, 2, 2, 4, 9, 5, 11, 29, 50, 14, 30, 110, 209, 285, 41, 82, 442, 1283, 1623, 1617, 122, 224, 1708, 8180, 16198, 12413, 9188, 365, 612, 6596, 49572, 167545, 203276, 95623, 52193, 1094, 1672, 25624, 302304, 1626073, 3401430, 2563481, 736757, 296511, 3281, 4568
Offset: 1

Views

Author

R. H. Hardin, Jan 25 2017

Keywords

Comments

Table starts
....1.......2.........4..........11.............30.............82
....2.......9........29.........110............442...........1708
....5......50.......209........1283...........8180..........49572
...14.....285......1623.......16198.........167545........1626073
...41....1617.....12413......203276........3401430.......52899445
..122....9188.....95623.....2563481.......69506779.....1732267694
..365...52193....736757....32354824.....1421127262....56764280423
.1094..296511...5678559...408458506....29066686772..1860912910152
.3281.1684466..43771933..5156857179...594539026170.61012156448915
.9842.9569425.337417047.65107404580.12161158312943

Examples

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

Links

Crossrefs

Column 1 is A007051(n-1).
Column 2 is A231413(n-1).
Row 1 is A021006(n-3).
Row 2 is A280853.

Formula

Empirical for column k:
k=1: a(n) = 4*a(n-1) -3*a(n-2)
k=2: a(n) = 6*a(n-1) -11*a(n-3) +4*a(n-4) for n>5
k=3: a(n) = 7*a(n-1) +10*a(n-2) -26*a(n-3) -64*a(n-4) -40*a(n-5) for n>6
k=4: [order 16] for n>18
k=5: [order 40] for n>42
Empirical for row n:
n=1: a(n) = 2*a(n-1) +2*a(n-2) for n>4
n=2: a(n) = 4*a(n-1) -2*a(n-2) +8*a(n-3) -8*a(n-4) for n>5
n=3: [order 13] for n>15
n=4: [order 55] for n>58

A204705 T(n,k) = Number of n X k 0..2 arrays with no occurrence of three equal elements in a row horizontally, vertically or nw-to-se diagonally, and new values 0..2 introduced in row major order.

Original entry on oeis.org

1, 2, 2, 4, 14, 4, 11, 96, 96, 11, 30, 726, 1417, 726, 30, 82, 5400, 22869, 22869, 5400, 82, 224, 40344, 362020, 835135, 362020, 40344, 224, 612, 301056, 5767683, 29846373, 29846373, 5767683, 301056, 612, 1672, 2247264, 91733605, 1074195735, 2389914827
Offset: 1

Views

Author

R. H. Hardin, Jan 18 2012

Keywords

Comments

Table starts
...1.......2..........4............11...............30..................82
...2......14.........96...........726.............5400...............40344
...4......96.......1417.........22869...........362020.............5767683
..11.....726......22869........835135.........29846373..........1074195735
..30....5400.....362020......29846373.......2389914827........192759130113
..82...40344....5767683....1074195735.....192759130113......34925753412578
.224..301056...91733605...38585324821...15521787839696....6319625580942184
.612.2247264.1459710274.1386815731346.1250634888842984.1144097011453760287

Examples

			Some solutions for n=5, k=3
..0..0..1....0..0..1....0..1..2....0..1..0....0..1..0....0..0..1....0..0..1
..1..0..0....2..0..1....1..1..2....2..1..2....0..1..1....1..2..2....1..0..2
..0..2..1....2..1..2....1..2..1....2..2..1....1..0..2....2..1..2....2..1..1
..0..0..2....1..2..1....0..1..2....0..1..1....1..2..2....2..2..0....0..1..2
..1..0..1....0..2..1....2..1..2....0..0..2....2..0..0....0..1..1....0..2..1

Links

Crossrefs

Column 1 is A021006(n-3).

A206650 T(n,k)=Number of nXk 0..2 arrays with no occurrence of three equal elements in a row horizontally or vertically, and new values 0..2 introduced in row major order.

Original entry on oeis.org

1, 2, 2, 4, 14, 4, 11, 96, 96, 11, 30, 726, 1625, 726, 30, 82, 5400, 30145, 30145, 5400, 82, 224, 40344, 545520, 1414023, 545520, 40344, 224, 612, 301056, 9937823, 64166392, 64166392, 9937823, 301056, 612, 1672, 2247264, 180716227, 2937839117
Offset: 1

Views

Author

R. H. Hardin Feb 11 2012

Keywords

Comments

Table starts
...1.......2..........4............11................30...................82
...2......14.........96...........726..............5400................40344
...4......96.......1625.........30145............545520..............9937823
..11.....726......30145.......1414023..........64166392...........2937839117
..30....5400.....545520......64166392........7241587334.........826407481102
..82...40344....9937823....2937839117......826407481102......235596375117548
.224..301056..180716227..134185561617....94026798615720....66923081500826797
.612.2247264.3287842038.6132878568838.10706810410657410.19028556027562892700

Examples

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

Links

Crossrefs

Column 1 is A021006(n-3)
Column 2 is A204699

A280859 T(n,k)=Number of nXk 0..2 arrays with no element equal to more than one of its king-move neighbors and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

1, 2, 2, 4, 9, 4, 11, 29, 29, 11, 30, 110, 46, 110, 30, 82, 442, 96, 96, 442, 82, 224, 1708, 256, 124, 256, 1708, 224, 612, 6596, 678, 216, 216, 678, 6596, 612, 1672, 25624, 1698, 462, 282, 462, 1698, 25624, 1672, 4568, 99432, 4358, 1005, 491, 491, 1005, 4358
Offset: 1

Views

Author

R. H. Hardin, Jan 09 2017

Keywords

Comments

Table starts
....1......2.....4...11...30...82..224...612..1672...4568...12480...34096
....2......9....29..110..442.1708.6596.25624.99432.385584.1495696.5802080
....4.....29....46...96..256..678.1698..4358.11218..28650...73354..188066
...11....110....96..124..216..462.1005..2010..3907...7756...15749...31804
...30....442...256..216..282..491..968..1857..3220...5281....8574...14339
...82...1708...678..462..491..712.1202..2268..4220...7108...11240...17330
..224...6596..1698.1005..968.1202.1738..2877..5244...9569...15872...24653
..612..25624..4358.2010.1857.2268.2877..4124..6722..12044...21713...35572
.1672..99432.11218.3907.3220.4220.5244..6722..9562..15351...27000...48087
.4568.385584.28650.7756.5281.7108.9569.12044.15351..21752...34547...59836

Examples

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

Links

Crossrefs

Column 1 is A021006(n-3).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-2) for n>4
k=2: a(n) = 4*a(n-1) -2*a(n-2) +8*a(n-3) -8*a(n-4) for n>5
k=3: a(n) = 2*a(n-1) +4*a(n-3) -a(n-4) +2*a(n-6) -2*a(n-7) for n>9
k=4: [order 12] for n>16
k=5: [order 12] for n>18
k=6: [order 16] for n>22
k=7: [order 23] for n>29

A204978 T(n,k)=Number of nXk 0..2 arrays with no occurrence of three equal elements in a row horizontally, vertically, diagonally or antidiagonally, and new values 0..2 introduced in row major order.

Original entry on oeis.org

1, 2, 2, 4, 14, 4, 11, 96, 96, 11, 30, 726, 1217, 726, 30, 82, 5400, 16813, 16813, 5400, 82, 224, 40344, 226908, 451468, 226908, 40344, 224, 612, 301056, 3074889, 11815256, 11815256, 3074889, 301056, 612, 1672, 2247264, 41618941, 311275169, 594430765
Offset: 1

Views

Author

R. H. Hardin Jan 21 2012

Keywords

Comments

Table starts
...1.......2.........4...........11.............30..............82
...2......14........96..........726...........5400...........40344
...4......96......1217........16813.........226908.........3074889
..11.....726.....16813.......451468.......11815256.......311275169
..30....5400....226908.....11815256......594430765.....30057359681
..82...40344...3074889....311275169....30057359681...2924321664147
.224..301056..41618941...8202482914..1519010368414.284567725886961
.612.2247264.563504926.216132492162.76757855124562

Examples

			Some solutions for n=5 k=3
..0..1..1....0..0..1....0..1..0....0..0..1....0..0..1....0..0..1....0..0..1
..0..1..2....2..0..0....2..1..2....2..0..0....0..0..1....0..0..1....1..2..1
..2..2..0....2..1..2....2..2..1....2..1..2....1..1..2....1..2..2....2..2..0
..0..1..1....1..1..2....0..1..1....0..1..2....2..1..1....1..2..2....0..0..2
..0..2..1....1..2..0....0..0..2....0..2..0....0..0..2....0..0..1....2..0..0

Links

Crossrefs

Column 1 is A021006(n-3)
Column 2 is A204699

A280961 T(n,k)=Number of nXk 0..2 arrays with no element equal to more than one of its horizontal, vertical and antidiagonal neighbors and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

1, 2, 2, 4, 9, 4, 11, 42, 42, 11, 30, 205, 241, 205, 30, 82, 997, 1554, 1554, 997, 82, 224, 4850, 9899, 14106, 9899, 4850, 224, 612, 23593, 63085, 126267, 126267, 63085, 23593, 612, 1672, 114769, 402077, 1121528, 1599234, 1121528, 402077, 114769, 1672, 4568
Offset: 1

Views

Author

R. H. Hardin, Jan 11 2017

Keywords

Comments

Table starts
....1.......2.........4.........11...........30............82...........224
....2.......9........42........205..........997..........4850.........23593
....4......42.......241.......1554.........9899.........63085........402077
...11.....205......1554......14106.......126267.......1121528.......9986376
...30.....997......9899.....126267......1599234......20029500.....251270618
...82....4850.....63085....1121528.....20029500.....346289502....6052621154
..224...23593....402077....9986376....251270618....6052621154..148137367471
..612..114769...2562733...88940022...3152475854..105777297118.3624578433814
.1672..558298..16334111..791997382..39530438497.1846875100200
.4568.2715861.104108376.7052519878.495721552160

Examples

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

Links

Crossrefs

Column 1 is A021006(n-3).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-2) for n>4
k=2: a(n) = 4*a(n-1) +4*a(n-2) +a(n-3) for n>4
k=3: [order 12] for n>16
k=4: [order 54] for n>57

A241130 T(n,k)=Number of nXk 0..2 arrays with no element equal to exactly two horizontal and vertical neighbors, with new values 0..2 introduced in row major order.

Original entry on oeis.org

1, 2, 2, 4, 9, 4, 11, 54, 54, 11, 30, 325, 723, 325, 30, 82, 1965, 9773, 9773, 1965, 82, 224, 11876, 132369, 295584, 132369, 11876, 224, 612, 71793, 1792237, 8974020, 8974020, 1792237, 71793, 612, 1672, 434007, 24269723, 272418756, 611547441, 272418756
Offset: 1

Views

Author

R. H. Hardin, Apr 16 2014

Keywords

Comments

Table starts
....1........2...........4..............11................30.................82
....2........9..........54.............325..............1965..............11876
....4.......54.........723............9773............132369............1792237
...11......325........9773..........295584...........8974020..........272418756
...30.....1965......132369.........8974020.........611547441........41661463219
...82....11876.....1792237.......272418756.......41661463219......6369134821236
..224....71793....24269723......8270609664.....2838552905280....973866422990563
..612...434007...328645291....251096788713...193403197619803.148910067386335603
.1672..2623694..4450319537...7623351508315.13177423185201303
.4568.15861001.60263615333.231446775293031

Examples

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

Links

Crossrefs

Column 1 is A021006(n-3)

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-2) for n>4
k=2: a(n) = 6*a(n-1) +2*a(n-2) -9*a(n-3) -10*a(n-4) +8*a(n-5) for n>6
k=3: [order 26]

A242472 T(n,k)=Number of length n+2 0..k arrays with no three equal elements in a row and new values 0..k introduced in 0..k order.

Original entry on oeis.org

3, 4, 5, 4, 11, 8, 4, 12, 30, 13, 4, 12, 40, 82, 21, 4, 12, 41, 143, 224, 34, 4, 12, 41, 158, 528, 612, 55, 4, 12, 41, 159, 663, 1979, 1672, 89, 4, 12, 41, 159, 684, 2944, 7466, 4568, 144, 4, 12, 41, 159, 685, 3204, 13537, 28246, 12480, 233, 4, 12, 41, 159, 685, 3232
Offset: 1

Views

Author

R. H. Hardin, May 15 2014

Keywords

Comments

Table starts
...3.....4......4.......4.......4.......4.......4.......4.......4.......4
...5....11.....12......12......12......12......12......12......12......12
...8....30.....40......41......41......41......41......41......41......41
..13....82....143.....158.....159.....159.....159.....159.....159.....159
..21...224....528.....663.....684.....685.....685.....685.....685.....685
..34...612...1979....2944....3204....3232....3233....3233....3233....3233
..55..1672...7466...13537...16042...16497...16533...16534...16534...16534
..89..4568..28246...63551...84412...90075...90817...90862...90863...90863
.144.12480.106992..301968..460174..520248..531812..532958..533013..533014
.233.34096.405481.1444795.2570411.3143900.3295779.3317613.3319308.3319374

Examples

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

Links

Crossrefs

Column 1 is A000045(n+3)
Column 2 is A021006(n-1)
Column 3 is A204678(n+2)
Column 4 is A222919(n+2)

Formula

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +2*a(n-2)
k=3: a(n) = 4*a(n-1) +a(n-2) -6*a(n-3) -3*a(n-4)
k=4: a(n) = 7*a(n-1) -7*a(n-2) -20*a(n-3) +10*a(n-4) +24*a(n-5) +8*a(n-6)
k=5: [order 8]
k=6: [order 10]
k=7: [order 12]
k=8: [order 14]

A280362 T(n,k)=Number of nXk 0..2 arrays with no element equal to more than one of its horizontal and vertical neighbors and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

1, 2, 2, 4, 9, 4, 11, 50, 50, 11, 30, 285, 571, 285, 30, 82, 1617, 6727, 6727, 1617, 82, 224, 9188, 78800, 164326, 78800, 9188, 224, 612, 52193, 924579, 3992071, 3992071, 924579, 52193, 612, 1672, 296511, 10844773, 97147710, 201054068, 97147710
Offset: 1

Views

Author

R. H. Hardin, Jan 01 2017

Keywords

Comments

Table starts
....1.......2...........4.............11...............30................82
....2.......9..........50............285.............1617..............9188
....4......50.........571...........6727............78800............924579
...11.....285........6727.........164326..........3992071..........97147710
...30....1617.......78800........3992071........201054068.......10144351660
...82....9188......924579.......97147710......10144351660.....1061399606602
..224...52193....10844773.....2363431872.....511688896140...111020011111453
..612..296511...127214104....57503068637...25812250379197.11613529410292966
.1672.1684466..1492251709..1399048151083.1302084964081326
.4568.9569425.17504551617.34038954765446

Examples

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

Links

Crossrefs

Column 1 is A021006(n-3).
Column 2 is A231413(n-1).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-2) for n>4
k=2: a(n) = 6*a(n-1) -11*a(n-3) +4*a(n-4) for n>5
k=3: [order 12]
k=4: [order 44]
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