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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A359024 Number of inequivalent tilings of an 8 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 21, 82, 861, 8038, 83609, 861159, 4495023
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

Column k = 8 of A227690.
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
Fixed: A000045, A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.
Free: A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.

A359025 Number of inequivalent tilings of a 9 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 30, 163, 2403, 32097, 459957, 6542578, 93604244
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

Column k = 9 of A227690.
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
Fixed: A000045, A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.
Free: A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.

A359026 Number of inequivalent tilings of a 10 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 51, 347, 7048, 130125, 2551794, 49828415
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

Column k = 10 of A227690.
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
Fixed: A000045, A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.
Free: A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.

A099233 Square array read by antidiagonals associated to sections of 1/(1-x-x^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 8, 1, 1, 1, 6, 15, 26, 28, 13, 1, 1, 1, 7, 21, 45, 69, 60, 21, 1, 1, 1, 8, 28, 71, 140, 181, 129, 34, 1, 1, 1, 9, 36, 105, 251, 431, 476, 277, 55, 1, 1, 1, 10, 45, 148, 413, 882, 1326, 1252, 595, 89, 1
Offset: 0

Views

Author

Paul Barry, Oct 08 2004

Keywords

Examples

			Rows begin
  1, 1, 1,  1,  1,   1, ...
  1, 1, 2,  3,  5,   8, ...
  1, 1, 3,  6, 13,  28, ...
  1, 1, 4, 10, 26,  69, ...
  1, 1, 5, 15, 45, 140, ...
Row 1 is the 0-section of 1/(1-x-x)   (A000079);
Row 2 is the 1-section of 1/(1-x-x^2) (A000045);
Row 3 is the 2-section of 1/(1-x-x^3) (A000930);
Row 4 is the 3-section of 1/(1-x-x^4) (A003269);
etc.

Links

Crossrefs

Sums of antidiagonals are A099236.
Columns include A000217, A008778.
Rows include A000045, A002478, A099234, A099235.
Main diagonal gives A099237.
Cf. A099238.

Formula

Square array T(n, k) = Sum_{j=0..n} binomial(k(n-j), j).
Rows are generated by 1/(1-x(1+x)^k) and satisfy a(n) = Sum_{k=0..n} binomial(n, k)a(n-k-1).

A189610 T(n,k)=Number of nXk array permutations with each element not moving, or moving one space E, S or NW.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 13, 20, 13, 1, 1, 28, 72, 72, 28, 1, 1, 60, 256, 464, 256, 60, 1, 1, 129, 912, 2853, 2853, 912, 129, 1, 1, 277, 3248, 17617, 30283, 17617, 3248, 277, 1, 1, 595, 11568, 108785, 321815, 321815, 108785, 11568, 595, 1, 1, 1278, 41200, 671452
Offset: 1

Views

Author

R. H. Hardin Apr 24 2011

Keywords

Comments

Table starts
.1....1......1........1..........1............1...............1
.1....3......6.......13.........28...........60.............129
.1....6.....20.......72........256..........912............3248
.1...13.....72......464.......2853........17617..........108785
.1...28....256.....2853......30283.......321815.........3414588
.1...60....912....17617.....321815......5897476.......107793872
.1..129...3248...108785....3414588....107793872......3394457868
.1..277..11568...671452...36212912...1968061359....106717857552
.1..595..41200..4144996..383990913..35917517449...3352763054744
.1.1278.146736.25586605.4071436782.655347656612.105288130659056

Examples

			Some solutions for 5X3
..0..5..1....4..0..1....4..1..2....0..5..1....0..1..2....0..1..2....0..5..1
..3..4..2....3..8..2....0..3..5....3..4..2....3..4..5....7..3..5....3..8..2
..6.11..7...10..7..5...10..6..7....6..7..8....6.11..7....6..4..8....6..4..7
.13..9..8....6..9.11....9.14..8....9.14.10...13.10..8....9.10.11....9.14.11
.12.10.14...12.13.14...12.13.11...12.13.11....9.12.14...12.13.14...12.10.13

Links

Crossrefs

Column 2 is A002478

A221693 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal or antidiagonal neighbor, without consecutive moves in the same direction.

Original entry on oeis.org

1, 3, 1, 6, 21, 1, 13, 152, 144, 1, 28, 1336, 3112, 987, 1, 60, 11036, 95076, 63676, 6765, 1, 129, 92660, 2627760, 7033429, 1302720, 46368, 1, 277, 774380, 74531185, 659020617, 520770016, 26650988, 317811, 1, 595, 6479664, 2098932856
Offset: 1

Views

Author

R. H. Hardin Jan 22 2013

Keywords

Comments

Table starts
.1.........3............6..............13................28................60
.1........21..........152............1336.............11036.............92660
.1.......144.........3112...........95076...........2627760..........74531185
.1.......987........63676.........7033429.........659020617.......64509677154
.1......6765......1302720.......520770016......164441398718....55433846144035
.1.....46368.....26650988.....38558774282....41023076707062.47681199145323119
.1....317811....545221260...2854945130275.10233875156794211
.1...2178309..11154026300.211383822013331
.1..14930352.228186749552
.1.102334155
.1

Examples

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

Links

Crossrefs

Column 2 is A033888
Row 1 is A002478

A221898 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal, vertical or antidiagonal neighbor, without move-in move-out straight through or left turns.

Original entry on oeis.org

1, 3, 3, 6, 31, 6, 13, 258, 266, 13, 28, 2241, 7667, 2305, 28, 60, 19408, 236532, 235332, 19936, 60, 129, 167960, 7235107, 25821731, 7197649, 172624, 129, 277, 1453952, 221465158, 2815434576, 2814706089, 220361275, 1494152, 277, 595, 12585637
Offset: 1

Views

Author

R. H. Hardin Jan 30 2013

Keywords

Comments

Table starts
...1........3..........6...........13............28...........60............129
...3.......31........258.........2241.........19408.......167960........1453952
...6......266.......7667.......236532.......7235107....221465158.....6778558415
..13.....2305.....235332.....25821731....2815434576.307126807101.33499434763000
..28....19936....7197649...2814706089.1093046005330
..60...172624..220361275.307131177266
.129..1494152.6744533268
.277.12933821
.595
Row and column 2, and row and column 3, have different values but the same recurrences

Examples

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

Links

Crossrefs

Column 1 and row 1 are A002478

A219741 T(n,k) = Unmatched value maps: number of nXk binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nXk array.

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 7, 13, 13, 7, 12, 28, 42, 28, 12, 21, 60, 126, 126, 60, 21, 37, 129, 387, 524, 387, 129, 37, 65, 277, 1180, 2229, 2229, 1180, 277, 65, 114, 595, 3606, 9425, 13322, 9425, 3606, 595, 114, 200, 1278, 11012, 39905, 78661, 78661, 39905, 11012, 1278, 200
Offset: 1

Views

Author

R. H. Hardin, Nov 26 2012

Keywords

Comments

Table starts
...1.....2......4........7.........12...........21.............37
...2.....6.....13.......28.........60..........129............277
...4....13.....42......126........387.........1180...........3606
...7....28....126......524.......2229.........9425..........39905
..12....60....387.....2229......13322........78661.........466288
..21...129...1180.....9425......78661.......647252........5350080
..37...277...3606....39905.....466288......5350080.......61758332
..65...595..11012...168925....2760690.....44159095......711479843
.114..1278..33636...715072...16350693....364647622.....8201909757
.200..2745.102733..3027049...96830726...3010723330....94531063074
.351..5896.313781.12813931..573456240..24858935864..1089590912023
.616.12664.958384.54243509.3396136349.205253857220.12558669019786

Examples

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

Links

Crossrefs

Column 1 is A005251(n+2).
Column 2 is A002478(n+1).
Column 3 is A105262(n+1) for n>1.
Main diagonal is A066864.
See A226444 for an array with very similar entries. - N. J. A. Sloane, Aug 22 2013

Formula

Zeilberger's Maple code (see links in A228285) would presumably give recurrences for the columns of this array. - N. J. A. Sloane, Aug 22 2013

A221929 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal, diagonal or antidiagonal neighbor, without move-in move-out straight through or left turns.

Original entry on oeis.org

1, 3, 1, 6, 25, 1, 13, 232, 184, 1, 28, 2197, 5748, 1367, 1, 60, 20933, 201378, 153652, 10179, 1, 129, 198589, 6888458, 20246283, 4053833, 75782, 1, 277, 1885324, 236114556, 2568601026, 2007184132, 107220027, 564169, 1, 595, 17896442
Offset: 1

Views

Author

R. H. Hardin Feb 01 2013

Keywords

Comments

Table starts
.1.......3..........6...........13...........28...........60............129
.1......25........232.........2197........20933.......198589........1885324
.1.....184.......5748.......201378......6888458....236114556.....8089630790
.1....1367.....153652.....20246283...2568601026.326363935235.41397744561659
.1...10179....4053833...2007184132.938516646422
.1...75782..107220027.199492714651
.1..564169.2834169558
.1.4200043
.1

Examples

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

Links

Crossrefs

Column 2 is A221777
Row 1 is A002478

A295918 T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 0 or 3 king-move neighboring 1s.

Original entry on oeis.org

2, 3, 3, 5, 6, 5, 8, 13, 13, 8, 13, 28, 39, 28, 13, 21, 60, 115, 115, 60, 21, 34, 129, 337, 467, 337, 129, 34, 55, 277, 993, 1880, 1880, 993, 277, 55, 89, 595, 2919, 7604, 10290, 7604, 2919, 595, 89, 144, 1278, 8587, 30721, 56955, 56955, 30721, 8587, 1278, 144, 233
Offset: 1

Views

Author

R. H. Hardin, Nov 29 2017

Keywords

Comments

Table starts
..2....3.....5......8......13........21.........34..........55............89
..3....6....13.....28......60.......129........277.........595..........1278
..5...13....39....115.....337.......993.......2919........8587.........25257
..8...28...115....467....1880......7604......30721......124117........501512
.13...60...337...1880...10290.....56955.....314044.....1732883.......9562608
.21..129...993...7604...56955....431844....3261576....24650278.....186318117
.34..277..2919..30721..314044...3261576...33703065...348555744....3605337986
.55..595..8587.124117.1732883..24650278..348555744..4933593439...69844332764
.89.1278.25257.501512.9562608.186318117.3605337986.69844332764.1353357158724

Examples

			Some solutions for n=5 k=4
..0..0..0..1. .0..0..1..0. .0..0..1..0. .0..0..0..0. .0..1..0..0
..1..0..0..0. .1..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..1
..0..0..0..1. .0..0..1..0. .1..0..0..1. .0..1..0..1. .1..0..0..0
..0..0..0..0. .1..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..1..0
..1..0..0..0. .0..0..0..1. .1..0..0..0. .0..0..0..0. .0..0..0..0

Links

Crossrefs

Column 1 is A000045(n+2).
Column 2 is A002478(n+1).

Formula

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: a(n) = 2*a(n-1) +3*a(n-2) -2*a(n-4)
k=4: a(n) = 2*a(n-1) +7*a(n-2) +6*a(n-3) -3*a(n-4) -4*a(n-5) +a(n-6)
k=5: [order 38]
k=6: [order 92]
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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