A084477 -id:A084477 - cp's OEIS frontend

A084480 Number of tilings of a 4 X 2n rectangle with L tetrominoes.

1, 2, 10, 42, 182, 790, 3432, 14914, 64814, 281680, 1224182, 5320310, 23122148, 100489226, 436727814, 1898026232, 8248853134, 35849651070, 155803171860, 677123141810, 2942788286798, 12789406189672, 55582969192486, 241564496305670, 1049843265359828

Offset: 0

Ralf Stephan, May 27 2003

Colin Barker, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 2.
Cristopher Moore, Some Polyomino Tilings of the Plane, arXiv:9905012 [math.CO], 1999.
Index entries for linear recurrences with constant coefficients, signature (4,2,-1,-4,-4,-2).

Cf. A084478, A084479, A084477, A084481, A174248, A226322, A232497.

LinearRecurrence[{4, 2, -1, -4, -4, -2}, {1, 2, 10, 42, 182, 790}, 25] (* Jean-François Alcover, Feb 25 2020 *)

Vec((1 - 2*x - x^3) / (1 - 4*x - 2*x^2 + x^3 + 4*x^4 + 4*x^5 + 2*x^6) + O(x^30)) \\ Colin Barker, Mar 28 2017

G.f.: (1-2*z-z^3) / (1-4*z-2*z^2+z^3+4*z^4+4*z^5+2*z^6).

a(n) = 4*a(n-1) + 2*a(n-2) - a(n-3) - 4*a(n-4) - 4*a(n-5) - 2*a(n-6) for n>5. - Colin Barker, Mar 28 2017

Inserted a(0)=1 by Alois P. Heinz, May 01 2013

A084478 Number of tilings of a 5 X 3n rectangle with right trominoes.

Original entry on oeis.org

1, 0, 72, 384, 8544, 76800, 1168512, 12785664, 170678784, 2014648320, 25633231872, 311423852544, 3892030055424, 47803588208640, 593425578949632, 7318730222874624, 90624271197041664, 1119402280975349760, 13847850677651745792, 171150049715628539904

Offset: 0

Ralf Stephan, May 27 2003

A right tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017
There is a sign typo with respect to the g.f. in the paper.
The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 2048, 0, 65536, 0,.. (tilings which have both vertical and horizontal faults), 0, 0, 64, 0, 0, 0, 0, 0.. (tilings which have horizontal but no vertical faults), 0, 0, 0, 0, 3136, 55296, 939008, 11649024... (tilings which have vertical faults but no horizontal faults), .. 1, 0, 8, 384, 3360, 21504, 163968 (essentially A084479) which have neither vertical nor horizontal faults. - R. J. Mathar, Dec 08 2022

Colin Barker, Table of n, a(n) for n = 0..900
D. Merlini, R. Sprugnoli, M. C. Verri, Strip tiling and regular grammars, Theo. Comp. Sci. 242 (1-2) (2000) 109-124, Proof of Theorem 4.2 (typo t^5 in the denominator of g.f. ought be t^6)
C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
Index entries for linear recurrences with constant coefficients, signature (2,103,280,380).

Cf. A046984, A084477, A084479 (INVERT transform), A084480, A084481,A351323, A351324, A236576 (straight trominoes), A233340 (mixed trominoes).

LinearRecurrence[{2, 103, 280, 380}, {72, 384, 8544, 76800}, 20] (* Jean-François Alcover, Jan 07 2019 *)

Vec(24*x^2*(3 + 10*x + 15*x^2) / (1 - 2*x - 103*x^2 - 280*x^3 - 380*x^4) + O(x^30)) \\ Colin Barker, Mar 27 2017

G.f.: (1 - 2*z - 31*z^2 - 40*z^3 - 20*z^4) / (1 - 2*z - 103*z^2 - 280*z^3 - 380*z^4).

a(n) = 2*a(n-1) + 103*a(n-2) + 280*a(n-3) + 380*a(n-4) for n > 4. - Colin Barker, Mar 27 2017

a(0) and a(1) prepended by Alois P. Heinz, Feb 21 2022

A046984 Number of ways to tile a 4 X 3n rectangle with right trominoes.

Original entry on oeis.org

1, 4, 18, 88, 468, 2672, 16072, 100064, 636368, 4097984, 26579488, 173093760, 1129796928, 7383588608, 48287978624, 315921649152, 2067346607360, 13530037877760, 88555066819072, 579620448450560, 3793872862974976, 24832858496561152, 162544900186359808

Offset: 0

Cristopher Moore (moore(AT)santafe.edu)

The sequence of tiling 2 X 3n rectangles with L-trominoes is 2^n. The sequence of tiling 3 X 2n rectangles is 2^n. All these tilings have vertical faults but no horizontal faults. - R. J. Mathar, Dec 08 2022

This sequence is the Hadamard sum of the following 4 sequences: 0, 0, 16, 64, 256, 1024, 4096... (A000302, tilings which have both vertical and horizontal faults), 0, 4, 0, 0, 0, 0, 0, ...(tilings which have horizontal but no vertical faults), 0, 0, 0, 16, 164, 1360, 10248, 73312, 508624, 3462592, 23291424.. (tilings which have vertical but no horizontal faults), 1, 0, 2, 8, 48, 288, 1728, 10368,.. (essentially A084477, tilings which have neither vertical nor horizontal faults). - R. J. Mathar, Dec 08 2022

Suggested on p. 96 of 1994 edition of "Polyominoes" by Samuel W. Golomb.

R. J. Mathar, Fault-free tilings with dominoes or trominoes.
Cristopher Moore, Preprint and figures
Cristopher Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
Index entries for linear recurrences with constant coefficients, signature (10,-22,-4).

Cf. A084478 (5 X 3n), A351323 (6 X n), A351324 (7 X 3n), A049086 (straight trominoes), A233339 (mixed trominoes).

a:= n-> (<<0|1|0>, <0|0|1>, <-4|-22|10>>^n. <<1, 4, 18>>)[1, 1]:
seq(a(n), n=0..22);  # Alois P. Heinz, Feb 21 2022

CoefficientList[Series[(1-6x)/(1-10x+22x^2+4x^3),{x,0,40}],x] (* or *) LinearRecurrence[{10,-22,-4},{1,4,18},40] (* Harvey P. Dale, Mar 31 2012 *)

a(n)=([0,1,0; 0,0,1; -4,-22,10]^n*[1;4;18])[1,1] \\ Charles R Greathouse IV, Feb 10 2017

G.f.: (1 - 6*x)/(1 - 10*x + 22*x^2 + 4*x^3).

a(0)=1, a(1)=4, a(2)=18, a(n)=10*a(n-1)-22*a(n-2)-4*a(n-3). - Harvey P. Dale, Mar 31 2012

A084479 Number of fault-free tilings of a 5 X 3n rectangle with right trominoes.

Original entry on oeis.org

72, 384, 3360, 21504, 163968, 1136640, 8283648, 58791936, 423121920, 3022872576, 21679875072, 155169515520, 1111792499712, 7961492434944, 57028930483200, 408439216748544, 2925470825868288, 20952944438968320, 150073631759459328, 1074876158496638976

Offset: 2

Ralf Stephan, May 27 2003

A tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A084478(n).

Colin Barker, Table of n, a(n) for n = 2..1000
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles, arXiv:math/0610925 [math.CO], 2006.
C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
Index entries for linear recurrences with constant coefficients, signature (2,31,40,20).

Cf. A084478, A084477, A084480, A084481.

LinearRecurrence[{2, 31, 40, 20}, {72, 384, 3360, 21504}, 20] (* Jean-François Alcover, Jan 07 2019 *)

Vec(24*x^2*(3 + 10*x + 15*x^2) / (1 - 2*x - 31*x^2 - 40*x^3 - 20*x^4) + O(x^30)) \\ Colin Barker, Mar 28 2017

G.f.: 24*z^2*(3 + 10*z + 15*z^2) / (1 - 2*z - 31*z^2 - 40*z^3 - 20*z^4).

a(n) = 2*a(n-1) + 31*a(n-2) + 40*a(n-3) + 20*a(n-4) for n > 5. - Colin Barker, Mar 28 2017

A084481 Number of fault-free tilings of a 4 X 2n rectangle with L tetrominoes.

Original entry on oeis.org

2, 6, 10, 18, 38, 84, 186, 410, 904, 1994, 4398, 9700, 21394, 47186, 104072, 229538, 506262, 1116596, 2462730, 5431722, 11980040, 26422810, 58277342, 128534724, 283492258, 625261858, 1379058440, 3041609138, 6708480134, 14796018708, 32633646554, 71975773242

Offset: 1

Ralf Stephan, May 27 2003

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A084480(n).

If the conjectured G.F. in A183304 is true, then a(n)= 2*A183304(n-1), n>3. - R. J. Mathar, Dec 02 2022

Colin Barker, Table of n, a(n) for n = 1..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
C. Moore, [math/9905012] Some Polyomino Tilings of the Plane
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Cf. A084478, A084479, A084480, A084477.

Vec(2*x*(1 + x)^2*(1 - x - x^3) / (1 - 2*x - x^3) + O(x^30)) \\ Colin Barker, Mar 28 2017

G.f.: 2*z*(1+z)^2*(1-z-z^3) / (1-2*z-z^3).

a(n) = 2*a(n-1) + a(n-3) for n>6. - Colin Barker, Mar 28 2017

A334396 Number of fault-free tilings of a 3 X n rectangle with squares and dominoes.

Original entry on oeis.org

0, 0, 2, 2, 10, 16, 52, 104, 286, 634, 1622, 3768, 9336, 22152, 54106, 129610, 314546, 756728, 1831196, 4413952, 10667462, 25735346, 62160046, 150020016, 362257392, 874442064, 2111291570, 5096782418, 12305249242, 29706645280, 71719568260

Offset: 1

Greg Dresden and Oluwatobi Jemima Alabi, Jul 06 2020

A fault-free tiling has no horizontal or vertical faults (that is to say, the tiling does not split along any interior horizontal or vertical line).

			a(4) = 2 because these are the only fault-free tilings of the 3 X 4 rectangle with squares and dominoes:
._ _ _ _     _ _ _ _
|_ _|_| |   | |_|_ _|
| |_ _|_|   |_|_ _| |
|_|_|_ _|   |_ _|_|_|

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-1).

Cf. A084477, A084479, A084481, A112577, A124997, A335747.

[n le 4 select 2*Floor((n-1)/2) else Self(n-1) +4*Self(n-2) -Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Jan 15 2022

a[n_]:= (2/3)*(Fibonacci[n-1, 2] - (-1)^n*Fibonacci[n-1]);
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 15 2022 *)

concat([0,0] , Vec(2*x^3/((1+x-x^2)*(1-2*x-x^2)) + O(x^30))) \\ Colin Barker, Aug 06 2020

[(2/3)*(lucas_number1(n-1,2,-1) - (-1)^n*lucas_number1(n-1,1,-1)) for n in (1..40)] # G. C. Greubel, Jan 15 2022

a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4) for n >= 5.
a(n) = 2*A112577(n-2) for n >= 2.
G.f.: 2*x^3 / ((1 + x - x^2)*(1 - 2*x - x^2)). - Colin Barker, Aug 06 2020

A270576 Expansion of g.f. (1+2x)/(1-6x).

Original entry on oeis.org

1, 8, 48, 288, 1728, 10368, 62208, 373248, 2239488, 13436928, 80621568, 483729408, 2902376448, 17414258688, 104485552128, 626913312768, 3761479876608, 22568879259648, 135413275557888, 812479653347328, 4874877920083968, 29249267520503808, 175495605123022848, 1052973630738137088

Offset: 0

Colin Barker, Mar 19 2016

Partial sums are 1, 9, 57, 345, 2073, 12441, ...

Essentially the same as A084477. - R. J. Mathar, Mar 21 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A000400 (powers of 6), A003949: (1+x)/(1-6*x), A084477.

PARI
```
Vec((1+2*x)/(1-6*x) + O(x^30))
```

G.f.: (1+2*x)/(1-6*x).
a(n) = 6*a(n-1) for n>1.
a(n) = 8*6^(n-1) for n>0.
E.g.f.: (4*exp(6*x) - 1)/3. - Elmo R. Oliveira, Mar 25 2025

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A084480 Number of tilings of a 4 X 2n rectangle with L tetrominoes.

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A084478 Number of tilings of a 5 X 3n rectangle with right trominoes.

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A046984 Number of ways to tile a 4 X 3n rectangle with right trominoes.

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A084479 Number of fault-free tilings of a 5 X 3n rectangle with right trominoes.

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A084481 Number of fault-free tilings of a 4 X 2n rectangle with L tetrominoes.

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A334396 Number of fault-free tilings of a 3 X n rectangle with squares and dominoes.

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A270576 Expansion of g.f. (1+2*x)/(1-6*x).

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A270576 Expansion of g.f. (1+2x)/(1-6x).