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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A054857 Number of ways to tile a 5 X n region with square tiles of size up to 5 X 5.

Original entry on oeis.org

1, 1, 8, 28, 117, 472, 1916, 7765, 31497, 127707, 517881, 2100025, 8515772, 34532063, 140030059, 567832091, 2302600696, 9337214060, 37863085664, 153537580071, 622606110920, 2524713292359, 10237896957896, 41515420557135
Offset: 0

Views

Author

Silvia Heubach (silvi(AT)cine.net), Apr 21 2000

Keywords

Examples

			a(2) = 8 as there is 1 tiling of a 5 X 2 region with only 1 X 1 tiles, 4 tilings with exactly one 2 X 2 tile and 3 tilings with exactly two 2 X 2 tiles.

Links

Crossrefs

Cf. A002478, A054856, A054858, A226548.
Column k=5 of A219924. - Alois P. Heinz, Dec 01 2012

Programs

  • Mathematica
    f[ {A_, B_} ] := Module[ {til = A, basic = B}, {Flatten[ Append[ til, ListConvolve[ A, B ] ] ], AppendTo[ basic, B[ [ -1 ] ] + B[ [ -2 ] ] + B[ [ -3 ] ] ]} ]; NumOfTilings[ n_ ] := Nest[ f, {{1, 1, 8, 28, 117, 472, 1916, 7765}, {1, 7, 13, 20, 35, 66, 118, 218}}, n - 2 ][ [ 1 ] ] NumOfTilings[ 30 ]

Formula

a(n) = b(1)a(n-1)+b(2)a(n-2)+...+b(n)a(0), a(0)=a(1)=1, b(n) as defined in A054858.
a(n) = 2*a(n-1) +7*a(n-2) +6*a(n-3) -a(n-4) -5*a(n-5) -2*a(n-6) -3*a(n-7) -a(n-8). - R. J. Mathar, Nov 02 2008
G.f.: -(x^3+x^2+x-1)/(x^8+3*x^7+2*x^6+5*x^5+x^4-6*x^3-7*x^2-2*x+1). - Colin Barker, Jul 10 2012

A264364 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,0 0,2 1,0 or -1,-2.

Original entry on oeis.org

1, 3, 1, 9, 6, 1, 18, 36, 13, 1, 36, 120, 169, 28, 1, 78, 400, 936, 784, 60, 1, 169, 1440, 5184, 7168, 3600, 129, 1, 364, 5184, 33408, 65536, 54720, 16641, 277, 1, 784, 18432, 215296, 730368, 831744, 418992, 76729, 595, 1, 1680, 65536, 1323792, 8139609
Offset: 1

Views

Author

R. H. Hardin, Nov 12 2015

Keywords

Comments

Table starts
.1....3.......9.........18...........36.............78.............169
.1....6......36........120..........400...........1440............5184
.1...13.....169........936.........5184..........33408..........215296
.1...28.....784.......7168........65536.........730368.........8139609
.1...60....3600......54720.......831744.......16066704.......310358689
.1..129...16641.....418992.....10549504......353333680.....11834176225
.1..277...76729....3204336....133818624.....7767356736....450847788304
.1..595..354025...24514000...1697440000...170773835200..17180991840016
.1.1278.1633284..187528608..21531453696..3754476071280.654674355426025
.1.2745.7535025.1434558960.273119121664.82543032602992

Examples

			Some solutions for n=4 k=4
..0..1..2..3..4....7..8..0..3..2....0..8..2..1..4....0..1..2..3..4
.12..6..7..8..9...12..1..5..6..4...12.13.14..3..7...12..6.14..8..7
..5.18.19.11.14...17.18.10.13..9....5..6.19.11..9....5.11.10.13..9
.10.16.24.13.17...22.11.24.16.14...10.16.24.18.17...15.23.24.16.19
.15.21.20.23.22...15.21.20.23.19...15.21.20.23.22...20.21.17.18.22

Links

Crossrefs

Column 2 is A002478(n+1).

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: a(n) = 3*a(n-1) +7*a(n-2) +3*a(n-3) -5*a(n-4) +3*a(n-5) -a(n-6)
k=4: a(n) = 3*a(n-1) +28*a(n-2) +57*a(n-3) +10*a(n-4) -24*a(n-5) +8*a(n-6)
k=5: a(n) = 11*a(n-1) +22*a(n-2) -8*a(n-3)
k=6: [order 30]
Empirical for row n:
n=1: a(n) = a(n-1) +3*a(n-3) +3*a(n-4) +3*a(n-5) +3*a(n-6) -2*a(n-8) -a(n-9)
n=2: a(n) = 3*a(n-1) +6*a(n-3) +4*a(n-4)

A359019 Number of inequivalent tilings of a 3 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 21, 39, 82, 163, 347, 717, 1533, 3232, 6927, 14748, 31645, 67690, 145322, 311535, 668997, 1435645, 3083301, 6619842, 14218066, 30533005, 65580338, 140847132, 302522253, 649759735, 1395611508, 2997573501, 6438470626, 13829057884, 29703388721, 63799607283, 137035047576, 294336860797, 632205714741
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Examples

			a(4) is 6 because of:
  +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
  | | | | |     | |   | | |   | | |   | | | | | |
  +-+-+-+ +     + +   +-+ +   +-+ +   +-+ +-+-+-+
  | | | | |     | |   | | |   | | |   | | |   | |
  +-+-+-+ +     + +-+-+-+ +-+-+-+ +-+-+-+ +   +-+
  | | | | |     | |   | | | |   | | | | | |   | |
  +-+-+-+ +-+-+-+ +   +-+ +-+   + +-+-+-+ +-+-+-+
  | | | | |     | |   | | | |   | | | | | | | | |
  +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+

Links

Crossrefs

Column k = 3 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.
Cf. A000930.

Formula

For n <= 1, a(n)=1;
otherwise for odd n > 1, a(n)=(A002478(n) + A000930(n) + 2 * A002478((n - 1) / 2) + 2 * A002478((n - 3) / 2)) / 4
and for even n, a(n)=(A002478(n) + A000930(n) + 2 * A002478((n - 2) / 2) + 2 * A002478(n / 2)) / 4
Alternatively, from Walter Trump:
For n <= 1, a(n)=1;
otherwise for odd n > 1, a(n)=(A000930(2n) + A000930(n) + 2 * A000930(n - 1) + 2 * A000930(n - 3)) / 4
and for even n, a(n)=(A000930(2n) + 2 * A000930(n - 2) + 3 * A000930(n)) / 4

A359020 Number of inequivalent tilings of a 4 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 4, 6, 13, 39, 115, 295, 861, 2403, 7048, 20377, 60008, 175978, 519589, 1532455, 4531277, 13395656, 39639758, 117301153, 347248981, 1028011708, 3043852214, 9012879842, 26689014028, 79033362580, 234045889421, 693101137571, 2052569508948
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Examples

			a(3) is 6 because of:
  +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
  | | | | |     | |   | | |   | | |   | | | | | |
  +-+-+-+ +     + +   +-+ +   +-+ +   +-+ +-+-+-+
  | | | | |     | |   | | |   | | |   | | |   | |
  +-+-+-+ +     + +-+-+-+ +-+-+-+ +-+-+-+ +   +-+
  | | | | |     | |   | | | |   | | | | | |   | |
  +-+-+-+ +-+-+-+ +   +-+ +-+   + +-+-+-+ +-+-+-+
  | | | | |     | |   | | | |   | | | | | | | | |
  +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+

Links

Crossrefs

Column k = 4 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.

Formula

For even n > 4
a(n) = (A054856(n) + compo(n) + 4 * A054856((n - 2) / 2) +
2 * A054856((n - 4) / 2) + 2 * A054856(n / 2) +
2 * Sum_{k=0..(n - 2) / 2} (A054856(k))) / 4
For odd n > 4
a(n) = (A054856(n) + compo(n) + 2 * A054856((n - 3) / 2) +
2 * A054856((n - 1) / 2) + 2 * Sum_ {k=0..(n - 3) / 2} (A054856(k))) / 4
Where compo(n) is the number of distinct compositions of n as a sum of 1, 2, (1+1) and 4.

A359021 Number of inequivalent tilings of a 5 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 5, 10, 39, 77, 521, 1985, 8038, 32097, 130125, 525676, 2131557, 8635656, 35017970, 141968455, 575692056, 2334344849, 9465939422, 38384559168, 155652202456, 631178976378, 2559476952229, 10378857744374, 42087027204278, 170665938023137, 692062856184512
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Examples

			a(2) is 5 because of:
  +-+-+ +-+-+ +-+-+ +-+-+ +-+-+
  | | | |   | |   | |   | |   |
  +-+-+ +-+-+ +   + +   + +-+-+
  | | | |   | |   | |   | |   |
  +-+-+ +   + +-+-+ +-+-+ +   +
  | | | |   | |   | | | | |   |
  +-+-+ +-+-+ +-+-+ +-+-+ +-+-+
  | | | |   | |   | | | | | | |
  +-+-+ +   + +   + +-+-+ +-+-+
  | | | |   | |   | | | | | | |
  +-+-+ +-+-+ +-+-+ +-+-+ +-+-+

Links

Crossrefs

Column k = 5 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.
Cf. A079975.

Formula

For even n > 5:
a(n) = (A054857(n) + A079975(n) + 2*A054857(n/2) + 2* fixed_md(n/2) + 2*A054857((n-4)/2) + 4*A054857((n-2)/2) + 2* (A054857((n/2)-1) + fixed_md((n/2)-1)))/4.
For odd n > 5:
a(n) = (A054857(n) + A079975(n) + 2*A054857((n-1)/2) + 4*A054857((n-3)/2) + 2*fixed_md((n-3)/2) + 2*A054857((n-5)/2) + 2*fixed_md((n-1)/2))/4.
where
fixed_md(1)=1, fixed_md(2)=3, fixed_md(3)=15 and for n > 3, fixed_md(n) = A054857(n-1) + A054857(n-2) + fixed_md(n-2)+ fixed_md(n-1) + 2*A054857(n-3) + fixed_md(n-3).

A220708 T(n,k)=Number of ways to reciprocally link elements of an nXk array either to themselves or to exactly two horizontal and antidiagonal neighbors, without consecutive collinear links.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 6, 5, 1, 1, 1, 8, 12, 13, 8, 1, 1, 1, 13, 24, 37, 28, 13, 1, 1, 1, 21, 48, 105, 107, 60, 21, 1, 1, 1, 34, 96, 298, 405, 317, 129, 34, 1, 1, 1, 55, 192, 846, 1520, 1617, 932, 277, 55, 1, 1, 1, 89, 384, 2404, 5706, 8338, 6412, 2749, 595
Offset: 1

Views

Author

R. H. Hardin Dec 18 2012

Keywords

Comments

Table starts
.1.1...1....1......1.......1.........1..........1............1.............1
.1.1...2....3......5.......8........13.........21...........34............55
.1.1...3....6.....12......24........48.........96..........192...........384
.1.1...5...13.....37.....105.......298........846.........2404..........6826
.1.1...8...28....107.....405......1520.......5706........21418.........80390
.1.1..13...60....317....1617......8338......42873.......221082.......1139020
.1.1..21..129....932....6412.....44976.....311193......2168111......15034974
.1.1..34..277...2749...25449....244029....2282408.....21692285.....204339688
.1.1..55..595...8101..101029...1322551...16699080....216083336....2751764573
.1.1..89.1278..23881..400986...7171769..122254180...2156218093...37166744488
.1.1.144.2745..70392.1591697..38885648..894932114..21512078437..501632637139
.1.1.233.5896.207497.6317904.210854845.6551231662.214655813486.6771890123950

Examples

			Some solutions for n=3 k=4 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10)
.00.67.47.00...00.00.00.00...00.00.67.47...00.00.00.00...00.67.47.00
.36.34.00.00...00.00.00.00...00.36.34.00...00.00.67.47...36.34.67.47
.00.00.00.00...00.00.00.00...00.00.00.00...00.36.34.00...00.36.34.00

Links

Crossrefs

Column 3 is A000045(n+1)
Column 4 is A002478
Row 2 is A000045
Row 3 is A003945(n-2)

A221937 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal or antidiagonal neighbor, without move-in move-out straight through or left turns.

Original entry on oeis.org

1, 3, 1, 6, 17, 1, 13, 105, 91, 1, 28, 783, 1599, 489, 1, 60, 5622, 36467, 24535, 2627, 1, 129, 40608, 799632, 1759041, 376389, 14113, 1, 277, 293084, 17595404, 119035112, 84547866, 5773962, 75819, 1, 595, 2115379, 387146025, 8118923949, 17626884659
Offset: 1

Views

Author

R. H. Hardin Feb 01 2013

Keywords

Comments

Table starts
.1........3...........6............13..............28...............60
.1.......17.........105...........783............5622............40608
.1.......91........1599.........36467..........799632.........17595404
.1......489.......24535.......1759041.......119035112.......8118923949
.1.....2627......376389......84547866.....17626884659....3719159201319
.1....14113.....5773962....4065176472...2611330803361.1705064784016177
.1....75819....88575493..195452177326.386840503637732
.1...407321..1358791848.9397301189681
.1..2188243.20844539836
.1.11755857
.1

Examples

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

Links

Crossrefs

Column 2 is A221731
Row 1 is A002478

A226444 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and L-tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 6, 5, 1, 1, 1, 1, 8, 13, 13, 8, 1, 1, 1, 1, 13, 28, 42, 28, 13, 1, 1, 1, 1, 21, 60, 126, 126, 60, 21, 1, 1, 1, 1, 34, 129, 387, 524, 387, 129, 34, 1, 1, 1, 1, 55, 277, 1180, 2229, 2229, 1180, 277, 55, 1, 1
Offset: 0

Views

Author

Alois P. Heinz, Jun 06 2013

Keywords

Comments

An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed.

Examples

			A(3,3) = 6:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|
  |_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|
  |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|.
Square array A(n,k) begins:
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 1,  2,   3,    5,     8,     13,      21,       34, ...
  1, 1,  3,   6,   13,    28,     60,     129,      277, ...
  1, 1,  5,  13,   42,   126,    387,    1180,     3606, ...
  1, 1,  8,  28,  126,   524,   2229,    9425,    39905, ...
  1, 1, 13,  60,  387,  2229,  13322,   78661,   466288, ...
  1, 1, 21, 129, 1180,  9425,  78661,  647252,  5350080, ...
  1, 1, 34, 277, 3606, 39905, 466288, 5350080, 61758332, ...

Links

Crossrefs

Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A002478, A105262, A219737(n-1) for n>2, A219738 (n-1) for n>2, A219739(n-1) for n>1, A219740(n-1) for n>2, A226543, A226544.
Main diagonal gives A066864(n-1).
See A219741 for an array with very similar entries. - N. J. A. Sloane, Aug 22 2013
Cf. A322494.

Programs

  • Maple
    b:= proc(n, l) option remember; local k, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; b(n, subsop(k=1, l))+
        `if`(k b(max(n, k), [0$min(n, k)]):
seq(seq(A(n, d-n), n=0..d), d=0..14);
[Zeilberger gives Maple code to find generating functions for the columns - see links in A228285. - N. J. A. Sloane, Aug 22 2013]
  • Mathematica
    b[n_, l_] := b[n, l] = Module[{k, t}, Which[Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 2}]], 0] ] ]; a[n_, k_] := b[Max[n, k], Array[0&, Min[n, k]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

Formula

The k-th column satisfies a recurrence of order Fibonacci(k+1) [Zeilberger] - see links in A228285. - N. J. A. Sloane, Aug 22 2013

A359022 Number of inequivalent tilings of a 6 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 9, 21, 115, 521, 1494, 15129, 83609, 459957, 2551794, 14150081, 78597739
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

Column k = 6 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.

A359023 Number of inequivalent tilings of a 7 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 12, 39, 295, 1985, 15129, 56978, 861159, 6542578, 49828415
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

Column k = 7 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.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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