Greg Dresden - cp's OEIS frontend

A386489 Expansion of (1-x)/((1+x+2x^2)(1-4*x+x^2)).

1, 2, 7, 30, 109, 402, 1511, 5638, 21021, 78474, 292887, 1093006, 4079181, 15223810, 56815879, 212039702, 791343293, 2953333114, 11021988791, 41134623134, 153516503405, 572931388658, 2138209053735, 7979904827430, 29781410249821, 111145736175722

Offset: 0

Greg Dresden and Madison Lingchen Zhou, Aug 20 2025

a(n) is the number of ways to tile a 2 X n board with squares, dominoes, and L-shaped quadrominoes. Here is one of the a(4)=109 possible tilings of a 2 X 4 board:
    _____
   | | |||
   ||____|
Compare to A030186 which counts the tilings with just squares and dominos.

Index entries for linear recurrences with constant coefficients, signature (3,1,7,-2).

Cf. A030186, A033505.

LinearRecurrence[{3, 1, 7, -2}, {1, 2, 7, 30}, 30]

a(n) = 3*a(n-1) + a(n-2) + 7*a(n-3) - 2*a(n-4).
a(n) = A030186(n) + 2*sum_{i=0..n-2}(A033505(n-i-3)*a(i) + A030186(n-i-3)*(a(i)+2*sum_{j=0..i} a(j)).
a(n) ~ (2 + sqrt(3))^(n+2) / (18 + 4*sqrt(3)). - Vaclav Kotesovec, Aug 21 2025
23*a(n) = -4*A001353(n)+13*A001353(n+1) +10*A001607(n+1)+8*A001607(n) . - R. J. Mathar, Aug 26 2025

A386222 Number of 3-dimensional tilings of a 2 X 2 X (n+1) box with the two upper right cells removed, using 2 X 2 X 1 plates and 1 X 2 X 1 dominos.

Original entry on oeis.org

1, 5, 34, 201, 1241, 7538, 46045, 280693, 1712338, 10443297, 63697825, 388506066, 2369604597, 14452808029, 88151396594, 537657790873, 3279312211305, 20001361622066, 121993408939853, 744068928339589, 4538266259447698, 27680043927136849, 168827650973959281

Offset: 0

Greg Dresden and Xiaoya Gao, Aug 13 2025

Here is the box for n=3:
    __________
   /   /   /   /|
  /_/___/_/ |__
 /   /   /   /| /   /|
/_/___/_/ |/_/ |
|   |   |   | /   /| /
|_|___|_|/_/ |/
|   |   |   |   |  /
|_|___|_|___| /.

			Here is one of the a(1)=5 ways to tile the shape for n=1, in this case with one flat plate on the bottom and one domino on top.
    ____
   /   /|
  /   / |____
 /   /  /   /|
/___/  /   / |
|   | /   /  /
|___|/___/  /
|       |  /
|_______| /.

Index entries for linear recurrences with constant coefficients, signature (5,9,-14).

Cf. A359884

LinearRecurrence[{5, 9, -14}, {1, 5, 34}, 30]

G.f.: 1/(1 - 5*x - 9*x^2 + 14*x^3).
a(n) = 5*a(n-1) + 9*a(n-2) - 14*a(n-3) for n >= 3.
a(n) = A359884(n) + 2*a(n-1).

A385242 Number of tilings of a 3 X n strip with dominos and U-shaped pentominos.

Original entry on oeis.org

1, 0, 3, 0, 12, 2, 50, 16, 210, 100, 888, 558, 3778, 2926, 16164, 14758, 69520, 72504, 300458, 349586, 1304390, 1662320, 5686114, 7821308, 24879632, 36497742, 109227706, 169207550, 480982532, 780370350, 2123682344, 3583760736, 9398963962, 16400994810, 41684827750

Offset: 0

Greg Dresden and Siting Jia, Jul 28 2025

Compare to A001835 which counts the tilings of a 3 X 2*(n-1) strip with just dominos. So, there will be 12 tilings of a 3 X 4 strip with dominos and U-shaped pentominos; 11 of them come from the U-free tilings counted in A001835(3), and here is the one additional tiling with two U's:
   _____
  |  |  |
  | |_| |
  |_|___|.

			Here are the a(5)=2 ways to tile the 3 X 5 strip with dominos and U's:
   _________     _________
  |___| |___|   | |  _  | |
  | | |_| | |   |_|_| |_|_|
  |_|_____|_|   |___|_|___|.

Index entries for linear recurrences with constant coefficients, signature (1,4,-3,0,-2).

Cf. A001835.

Join[{1}, LinearRecurrence[{1, 4, -3, 0, -2}, {0, 3, 0, 12, 2}, 40]]

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

a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 2*a(n-5) for n >= 6.

A385868 Number of ways to tile a hexagonal strip made up of n equilateral triangles, using triangles, diamonds, and trapezoids.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 39, 66, 110, 200, 604, 1032, 1741, 3149, 9476, 16202, 27337, 49461, 148841, 254466, 429308, 776774, 2337580, 3996430, 6742361, 12199339, 36711974, 62764458, 105889743, 191592331, 576566591, 985724436, 1663012914, 3008983882, 9055057632, 15480937786

Offset: 0

Greg Dresden and Sean Choi, Jul 10 2025

Here is the hexagonal strip:
    ______________      __
   /\  /\  /\  /\  /      \  /
  /\/\/\/\/  ...   \/
  \  /\  /\  /\  /\        /\
   \/\/\/\/\      /\
The three types of tiles are triangles, diamonds, and trapezoids (each of which can be rotated). Here are the three types of tiles:
  __        __             __
  \  /        \   \           /    \
   \/   and    \_\   and   /____\.
Compare to A356622 and A356623, which are on a similar board but only use triangles and diamonds.

			Here is one of the a(13) = 3149 possible tilings for this strip of 13 triangular cells:
   ____________
  /   /\   \   \
 /__ /__\   \ __\
 \      /\  /\
  \____/__\/__\.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,15,0,0,0,9,0,0,0,32,0,0,0,9,0,0,0,1,0,0,0,-1).

Cf. A356623, A356623.

a[0] = 1; a[1] = 1; a[2] = 2; a[3] = 4; a[4] = 7; a[5] = 13; a[6] = 39; a[7] = 66;
a[n_] := a[n] = Switch[Mod[n, 4],
    0, a[n-1] + a[n-3] + 2 a[n-4] + 3 a[n-5] + 2 a[n-6] + a[n-7],
    1, a[n-1] + a[n-2] + a[n-4] + a[n-5] + a[n-6],
    2, a[n-1] + a[n-2] + 3 a[n-3] + a[n-4] + 3 a[n-5] + 2 a[n-6] + a[n-7],
    3, a[n-1] + a[n-2] + a[n-3] + a[n-4] + a[n-5] + a[n-6]];
Table[a[n], {n, 0, 40}]

a(n) = 15*a(n-4) + 9*a(n-8) + 32*a(n-12) + 9*a(n-16) + 1*a(n-20)  - 1*a(n-24).
a(4*n+2) = t(2*n+1)^2 + t(2*n)^2 + 2*t(2*n)*t(2*n-1) + a(4*n) + 2*a(4*n-1) + Sum_{i=1..n-1} a(4*i)*(t(2*(n-i))^2 + 2*t(2*(n-i))*t(2*(n-i)-1)) + Sum_{i=1..n-1} a(4*i-1)*2*t(2*(n-i))^2, for t(n) = A000073(n+2).
G.f.: (x^18-x^17+x^16-x^14+4*x^12-6*x^11-x^10+4*x^9+4*x^8-6*x^7-9*x^6+2*x^5+8*x^4-4*x^3 -2*x^2-x-1) / (-x^24+x^20+9*x^16+32*x^12+9*x^8+15*x^4-1). - Alois P. Heinz, Jul 21 2025

A385933 Number of ways to tile a "central bump" strip of length n with 1 X 1 squares and 1 X 3 rectangles.

Original entry on oeis.org

4, 9, 13, 25, 30, 35, 52, 78, 121, 189, 271, 388, 561, 812, 1204, 1785, 2617, 3837, 5602, 8179, 12000, 17606, 25825, 37881, 55483, 81264, 119089, 174520, 255828, 375017, 549589, 805425, 1180342, 1729779, 2535196, 3715630, 5445561, 7980917, 11696455, 17141772

Offset: 0

Greg Dresden and Saim Usmani, Jul 12 2025

nonn

a(n) is the number of ways to tile this "central bump" strip of length n (shown here at n=18) with 1 X 1 squares and 1 X 3 rectangles which can be horizontal or vertical:
                   _
                 ||_
 _____________|_|||_____________
|||_|||_|||_|||_|||_|||_|
                |||_|
                  |_|

			For n = 0 there is no horizontal strip but there is still the "central bump". Here are the a(n) = 4 ways to tile this (disjoint) structure with 1 X 1 squares and 1 X 3 rectangles.
   _           _           _           _
 _|_|_       _|_|_       _|_|_       _|_|_
|_|_|_|     |_____|     |_|_|_|     |_____|
 _____       _____       _____       _____
|_|_|_|     |_|_|_|     |_____|     |_____|
  |_|         |_|         |_|         |_|

Cf. A000930.

LinearRecurrence[{1,0,1,-1,1,1,0,0,-1},{4,9,13,25,30,35,52,78,121},61]

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) + a(n-6) - a(n-9).
a(2*n) = a(2*n-2) + a(2*n-4) + 2*a(2*n-6) + a(2*n-7) + a(2*n-8).
a(2*n+1) = a(2*n-1) + 2*a(2*n-4) + a(2*n-5) + 2*a(2*n-6).
a(2*n+3) = 25*b(n)^2 + 26*b(n)*b(n-2) + 10*b(n)*b(n-1) + 9*b(n-2)^2 + 8*b(n-1)*b(n-2) for b(n) = A000930(n) the Narayana Cow sequence.
G.f.: (4 + 5*x + 4*x^2 + 8*x^3 - 3*x^5 - 8*x^6 - x^7)/(1 - x - x^3 + x^4 - x^5 - x^6 + x^9).

A385717 a(n) = a(n-1) + a(n-2) + a(n-3), with a(1) = 4, a(2) = 13, a(3) = 42.

Original entry on oeis.org

4, 13, 42, 59, 114, 215, 388, 717, 1320, 2425, 4462, 8207, 15094, 27763, 51064, 93921, 172748, 317733, 584402, 1074883, 1977018, 3636303, 6688204, 12301525, 22626032, 41615761, 76543318, 140785111, 258944190, 476272619

Offset: 1

Greg Dresden and Jiarui Zhou, Jul 07 2025

For n >= 3, a(n) is the number of ways to tile this shape of length n with 1 X 1 squares, 1 X 2 dominos, and 1 X 3 trominos:
   _
 ||_|_______
|||_|||_|||
  |||
As an example, here is one of the a(8) = 717 ways to tile this shape of length 8:
   _
 | ||_________
|| | ||_|___|
  |||

Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A100683, A354080.

LinearRecurrence[{1, 1, 1}, {4, 13, 42}, 30]

a(n) = 9*A354080(n-2) + 2*A100683(n) for n >= 2.

G.f.: x*(4 + 9*x + 25*x^2)/(1 - x - x^2 - x^3). - Stefano Spezia, Jul 08 2025

A385633 a(n) = a(n-1) + a(n-3), with a(0) = 1, a(1) = 4, a(2) = 8.

Original entry on oeis.org

1, 4, 8, 9, 13, 21, 30, 43, 64, 94, 137, 201, 295, 432, 633, 928, 1360, 1993, 2921, 4281, 6274, 9195, 13476, 19750, 28945, 42421, 62171, 91116, 133537, 195708, 286824, 420361, 616069, 902893, 1323254, 1939323, 2842216, 4165470, 6104793, 8947009, 13112479, 19217272

Offset: 0

Greg Dresden and Saim Usmani, Jul 05 2025

a(n) is the number of ways to tile this shape, of length n, with 1 X 1 squares and 1 X 3 rectangles (which can be either horizontal or vertical).
    _
   ||
   |||______________
   |||_|||_|||_|
   |||
   |_|

			Shown here is one of the a(4)=13 ways to tile this shape of length 4:
   _
  | |_
  | | |____
  |_| |_|_|
  |_|_|
  |_|

Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000930, A179070.

LinearRecurrence[{1, 0, 1}, {1, 4, 8}, 40]

a(n) = b(n+4) + 2*b(n-5) for b(n) = A000930(n), Narayana's cows sequence.

G.f.: (1 + 3*x + 4*x^2)/(1 - x - x^3). - Stefano Spezia, Jul 06 2025

A376031 Number of ways to tile a 3 x (2*n) rectangle with dominoes and T's.

Original entry on oeis.org

1, 3, 18, 112, 692, 4294, 26624, 165086, 1023662, 6347440, 39358774, 244053158, 1513307844, 9383614226, 58185263358, 360791140032, 2237168644134, 13872079956206, 86017029971684, 533368425534858, 3307273890427894, 20507514248408832, 127161570097317790

Offset: 0

Greg Dresden and Lucas MingQu Xia, Sep 06 2024

a(n) is the number of ways to tile a 3 X (2*n) rectangle with two kinds of tiles: dominoes (made up of 2 cells) and T's (made up of 4 cells), each of which can be rotated as needed.

			For n=3, here is one of the a(3) = 112 ways to tile a 3 x 6 rectangle using our dominoes and T's:
 ___________
| |___| | | |
|  _|_  |_|_|
|_|___|_|___|.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,7,4,-8,2).

Cf. A033506, A219968, A337492, A346054.

LinearRecurrence[{5, 7, 4, -8, 2}, {1, 3, 18, 112, 692}, 30] (* Paolo Xausa, Sep 23 2024 *)

a(n) = 5*a(n-1) + 7*a(n-2) + 4*a(n-3) - 8*a(n-4) + 2*a(n-5).

G.f.: -(2*x^4-3*x^3-4*x^2-2*x+1)/(2*x^5-8*x^4+4*x^3+7*x^2+5*x-1).

A375823 Number of ways to tile a 3-row trapezoid of average length n with triangular and rectangular tiles, each of size 3.

Original entry on oeis.org

0, 1, 3, 6, 16, 43, 107, 271, 695, 1769, 4499, 11464, 29202, 74360, 189382, 482339, 1228417, 3128538, 7967848, 20292665, 51681683, 131623881, 335222157, 853749843, 2174345679, 5537663440, 14103422412, 35918853816, 91478793556, 232979863477, 593357374127

Offset: 0

Greg Dresden and Mingjun Oliver Ouyang, Aug 30 2024

Here is the 3-row trapezoid of average length 6 (with 18 cells):
     _ ___ _ ___ _
    |   |   |   |   |   |
   |__|___|_|___| |_
  |   |   |   |   |   |   |
 |__|___|_|___| |___|_
|   |   |   |   |   |   |   |
|_|___|_|___|_|___|_|,
and here are the two types of (triangular and rectangular) tiles of size 3, which can be rotated as needed:
   _
  |   |
 |__|_     _________
|   |   |   |   |   |   |
|_|___|,  |_|___|_|.
As an example, here is one of the a(6) = 107 ways to tile the 3-row trapezoid
     _ ___ _________
    |   |   |           |
   |  |_  |_________|_
  |   |   |   |       |   |
 |  |   |  |     |   |
|   |       |   |   |       |
|_|_______|_|___|_____|.

Index entries for linear recurrences with constant coefficients, signature (2,0,4,-1,0,-1).

Cf. A077939, A077961, A375821.

LinearRecurrence[{2, 0, 4, -1, 0, -1}, {0, 1, 3, 6, 16, 43}, 40]

a(n) = 2*a(n-1) + 4*a(n-3) - a(n-4) - a(n-6).
G.f.: x*(1 + x)/((1 + x^2 - x^3)*(1 - 2*x - x^2 - x^3)).
a(n) = (A077939(n) - A077961(n))/2.

A375821 Number of ways to tile a 3-row parallelogram of length n with triangular and rectangular tiles, each of size 3.

Original entry on oeis.org

1, 1, 2, 7, 17, 41, 107, 274, 693, 1766, 4504, 11465, 29194, 74364, 189391, 482327, 1228412, 3128559, 7967841, 20292639, 51681711, 131623900, 335222103, 853749852, 2174345752, 5537663377, 14103422348, 35918853952, 91478793557, 232979863277, 593357374262

Offset: 0

Greg Dresden and Mingjun Oliver Ouyang, Aug 30 2024

Here is the 3-row parallelogram of length 6 (with 18 cells):
     _ ___ _ ___ _ ___
    |   |   |   |   |   |   |
   |__|___|_|___| |___|
  |   |   |   |   |   |   |
 |__|___|_|___| |___|
|   |   |   |   |   |   |
|_|___|_|___|_|___|,
and here are the two types of (triangular and rectangular) tiles of size 3, which can be rotated as needed:
   _
  |   |
 |__|_     _________
|   |   |   |   |   |   |
|_|___|,  |_|___|_|.
As an example, here is one of the a(6) = 107 ways to tile the 3 x 6 parallelogram:
     _ _______ _________
    |   |       |           |
   |  |_     |__________|
  |   |   |   |           |
 |  |   |_|___________|
|   |       |           |
|_|_______|_________|.

Index entries for linear recurrences with constant coefficients, signature (2,0,4,-1,0,-1).

Cf. A077939, A077961, A375823.

LinearRecurrence[{2, 0, 4, -1, 0, -1}, {1, 1, 2, 7, 17, 41}, 40]

a(n) = 2*a(n-1) + 4*a(n-3) - a(n-4) - a(n-6).
G.f.: (1 - x - x^3)/((1 + x^2 - x^3)*(1 - 2*x - x^2 - x^3)).
a(n) = (A077939(n) + A077961(n))/2.

cp's OEIS Frontend

User: Greg Dresden

A386489 Expansion of (1-x)/((1+x+2*x^2)*(1-4*x+x^2)).

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A386222 Number of 3-dimensional tilings of a 2 X 2 X (n+1) box with the two upper right cells removed, using 2 X 2 X 1 plates and 1 X 2 X 1 dominos.

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A385242 Number of tilings of a 3 X n strip with dominos and U-shaped pentominos.

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A385868 Number of ways to tile a hexagonal strip made up of n equilateral triangles, using triangles, diamonds, and trapezoids.

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A385933 Number of ways to tile a "central bump" strip of length n with 1 X 1 squares and 1 X 3 rectangles.

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A385717 a(n) = a(n-1) + a(n-2) + a(n-3), with a(1) = 4, a(2) = 13, a(3) = 42.

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A385633 a(n) = a(n-1) + a(n-3), with a(0) = 1, a(1) = 4, a(2) = 8.

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A376031 Number of ways to tile a 3 x (2*n) rectangle with dominoes and T's.

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A386489 Expansion of (1-x)/((1+x+2x^2)(1-4*x+x^2)).