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.

Showing 1-9 of 9 results.

A359884 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates and 1 X 2 X 1 dominos.

Original entry on oeis.org

1, 3, 24, 133, 839, 5056, 30969, 188603, 1150952, 7018621, 42811231, 261110416, 1592592465, 9713598835, 59245780536, 361354997685, 2203996629559, 13442737199456, 81990685695721, 500082110459883, 3050128402768520, 18603511408241453, 113467563119685583
Offset: 0

Views

Author

Gerhard Kirchner, Jan 20 2023

Keywords

Comments

The first recurrence is derived in "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 4.

Examples

			a(1) = 3
      _______         _______          _______
    /       /|      /   /   /|       /______ /|
   /______ / |     /__ /__ / |      /______ /||
   |       | /     |   |   | /      |       ||/
   |_______|/      |___|___|/       |_______|/

Links

Crossrefs

Cf. A006253, A001045, A335559, A359885, A359886.

Programs

  • Mathematica
    LinearRecurrence[{5, 9, -14}, {1, 3, 24}, 25] (* Paolo Xausa, Jun 24 2024 *)
  • Maxima
    /* See link "Maxima code". */

Formula

G.f.: (1 - 2*x) / (1 - 5*x - 9*x^2 + 14*x^3).
a(n) = 3*a(n-1) + c(n-1) + 7*a(n-2) where c(n) = 8*a(n-1) + 2*c(n-1) with a(n),c(n) <= 0 for n <= 0 except for a(0)=1.
a(n) = 5*a(n-1) + 9*a(n-2) - 14*a(n-3) for n >= 3.

A360064 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 5, 89, 1177, 16873, 237977, 3366793, 47599097, 673035625, 9516252633, 134553882441, 1902506043833, 26900227288361, 380352114739609, 5377937177440009, 76040613721296249, 1075165950495479017, 15202163218500810073, 214948926180739194569
Offset: 0

Views

Author

Gerhard Kirchner, Jan 30 2023

Keywords

Comments

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 8.

Examples

			4 rotations:
   ___ ___     ___ ___
  |   |   |   |   |   | (cross sections)
  |   |___|   |___|___|
  |       |   |   |   |
  |_______|   |___|___| a(1) = 4 + 1 = 5.

Links

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359885, A359886, A360065, A360066.

Programs

  • Mathematica
    LinearRecurrence[{13, 20, -64, 112, 224, -128}, {1, 5, 89, 1177, 16873, 237977}, 25] (* Paolo Xausa, Oct 02 2024 *)

Formula

G.f.: (1 - 8*x + 4*x^2 - 16*x^3) / (1 - 13*x - 20*x^2 + 64*x^3 - 112*x^4 - 224*x^5 + 128*x^6).
Recurrence 1:
a(n) = 5*a(n-1) + 2*b(n-1) + c(n-1) + d(n-1) + e(n-1) + 8*a(n-2) + 4*b(n-2) + c(n-2) + 2*d(n-2),
b(n) = 8*a(n-1) + 4*b(n-1) + 2*c(n-1),
c(n) = 20*a(n-1) + 6*b(n-1) + 4*c(n-1) + 4*d(n-1) + 2*e(n-1),
d(n) = 4*a(n-1), e(n) = 16*a(n-1) + 4*b(n-1),
with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.
Recurrence 2:
a(n) = 13*a(n-1) + 20*a(n-2) - 64*a(n-3) + 112*a(n-4) + 224*a(n-5) - 128*a(n-6) for n >= 6. For n < 6, recurrence 1 can be used.

A360065 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 2, 45, 412, 4705, 50374, 549109, 5955544, 64683649, 702259786, 7625147293, 82791470836, 898931464993, 9760376329678, 105975828745957, 1150659965697328, 12493588746237697, 135652375422278290, 1472880803124594061, 15992184812239930060, 173639288800074705121
Offset: 0

Views

Author

Gerhard Kirchner, Jan 30 2023

Keywords

Comments

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 9.

Examples

			a(2)=45
1) Two parallel trominos and one domino: There are 3 middle axes of the 2 X 2 cube with 4 rotation images each: 12 images.
       ___             ___         ___ ___
     /__ /|          /   /|      /__ /   /|
   /__ /| |___     /__ /  |    /__ /__ /  |
  |   | |/__ /|   |   |  /    |   |   |  /|
  |   |/__ /| | + |___|/   =  |   |___|/| |
  |       | |/                |       | |/
  |_______|/                  |_______|/
2) Two "linked" trominos and one domino: 12 rotation images and, as there is no symmetry plane, 12 mirror images: 24 images.
       ___                       ___         ___ ___
     /   /|                    /   /|      /   /   /|
   /__ /  |      _______     /__ /  |    /__ /__ /  |
  |   |  /     /__     /|   |   |  /    |   |   |  /|
  |   | |  +  |  /__ /  | + |___|/   =  |   |___|/  |
  |   | |     |_|   |  /                |   |   |  /
  |___|/        |___|/                  |___|___|/
3) Using only dominos: A006253(2)=9 ways, Sum: a(2) = 12 + 24 + 9 = 45.

Links

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360066.

Programs

  • Mathematica
    LinearRecurrence[{7, 42, 6, -81, 27}, {1, 2, 45, 412, 4705}, 25] (* Paolo Xausa, Oct 02 2024 *)

Formula

G.f.: (1 - 5*x - 11*x^2 + 7*x^3) / (1 - 7*x - 42*x^2 - 6*x^3 + 81*x^4 - 27*x^5).
Recurrence 1:
a(n) = 2*a(n-1) + b(n-1) + c(n-1) + 13*a(n-2) + 2*b(n-2) + c(n-2) + 2*d(n-2),
b(n) = 12*a(n-1) + 2*b(n-1) + 2*c(n-1) + e(n-1),
c(n) = 16*a(n-1) + 6*b(n-1) + c(n-1) + 2*e(n-1),
d(n) = 4*a(n-1) + 2*b(n-1) + d(n-1),
e(n) = 16*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1),
with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.
Recurrence 2:
a(n) = 7*a(n-1) + 42*a(n-2) + 6*a(n-3) - 81*a(n-4) + 27*a(n-5) for n >= 5.
For n < 5, recurrence 1 can be used.

A359886 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates and trominos (three 1 X 1 X 1 cubes).

Original entry on oeis.org

1, 1, 3, 49, 231, 789, 4771, 27225, 122799, 607469, 3255979, 16253649, 80098519, 409480005, 2079921395, 10411734921, 52523676351, 266059774429, 1341128940795, 6758479842689, 34138205819239, 172324729379509, 869131661400259, 4386075013348025, 22138673661637327
Offset: 0

Views

Author

Gerhard Kirchner, Jan 20 2023

Keywords

Comments

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 6.

Examples

			a(3) = 49.
The number of tilings only using plates is A001045(3) = 5.
The number of tilings only using trominos is A359885(1) = 44.
These terms are to be added as, for n=3, there is no tiling using both tiles.

Links

Crossrefs

Cf. A006253, A001045, A335559, A359884, A359885.

Programs

  • Mathematica
    LinearRecurrence[{2, 1, 58, 72, 32, -128}, {1, 1, 3, 49, 231, 789}, 30] (* Paolo Xausa, Jun 24 2024 *)
  • Maxima
    /* See link "Maxima code". */

Formula

G.f.: (1 - x - 16*x^3) / (1 - 2*x - x^2 - 58*x^3 - 72*x^4 - 32*x^5 + 128*x^6).
Recurrence 1:
a(n) = a(n-1) + 3*c(n-2) + 2*a(n-2) + 4*c(n-3) + 8*a(n-3),
c(n) = 12*a(n-1) + c(n-1) + 16*a(n-2) + 16*c(n-3),
with a(n),c(n) <= 0 for n <= 0 except for a(0)=1.
Recurrence 2:
a(n) = 2*a(n-1) + a(n-2) + 58*a(n-3) + 72*a(n-4) + 32*a(n-5) - 128*a(n-6) for n >= 6.
For n < 6, recurrence 1 can be used.

A360066 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 11, 444, 13311, 422617, 13265660, 417336617, 13123557903, 412719195520, 12979269602143, 408175860119021, 12836425011761592, 403683424226081169, 12695147020245034099, 399240466722076292612, 12555423726269799691295, 394846409914451855949249
Offset: 0

Views

Author

Gerhard Kirchner, Jan 30 2023

Keywords

Comments

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 10.

Links

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065.

Programs

Formula

G.f.: (1 - 15*x - 18*x^2 - 23*x^3 + 7*x^4) / (1 - 26*x - 176*x^2 + 146*x^3 + 14*x^4 + 140*x^5 - 27*x^6).
Recurrence 1:
a(n) = 11*a(n-1) + 4*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 29*a(n-2) + 6*b(n-2) + c(n-2) + 2*d(n-2),
b(n) = 32*a(n-1) + 9*b(n-1) + 4*c(n-1) + 2*d(n-1) + e(n-1),
c(n) = 52*a(n-1) + 14*b(n-1) + 5*c(n-1) + 4*d(n-1) + 2*e(n-1),
d(n) = 14*a(n-1) + 3*b(n-1) + d(n-1),
e(n) = 48*a(n-1) + 11*b(n-1) + 2*c(n-1) + 2*d(n-1),
with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.
Recurrence 2:
a(n) = 26*a(n-1) + 176*a(n-2) - 146*a(n-3) - 14*a(n-4) - 140*a(n-5) + 27*a(n-6) for n >= 6. For n < 6, recurrence 1 can be used.

A360575 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 1 X 1 dominos and 2 X 2 X 1 plates.

Original entry on oeis.org

1, 8, 153, 2470, 41571, 693850, 11602579, 193942076, 3242104149, 54196828452, 905988148597, 15145052657186, 253174020910071, 4232212575080006, 70748267813548207, 1182671546039152712, 19770264765434877913, 330491902143708738464
Offset: 0

Views

Author

Gerhard Kirchner, Feb 12 2023

Keywords

Comments

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 11.

Links

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360576, A360577.

Formula

G.f.: (1-8*x+4*x^2+11*x^3-6*x^4) / (1-16*x-21*x^2+157*x^3-100*x^4-65*x^5+42*x^6).
Recurrence 1:
a(n) = 8*a(n-1) + 3*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 7*a(n-2)
b(n) = 12*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1) + e(n-1)
c(n) = 16*a(n-1) + 4*b(n-1) + 2*c(n-1)
d(n) = 2*a(n-1) + b(n-1) + d(n-1)
e(n) = 12*a(n-1) + 3*b(n-1)
with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
Recurrence 2:
a(n)=16*a(n-1) + 21*a(n-2) - 157*a(n-3) + 100*a(n-4) + 65*a(n-5) - 42*a(n-6)
for n>=6. For n<6, recurrence 1 can be used.

A360576 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 2 X 1 plates and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 6, 122, 1768, 28844, 457592, 7318760, 116806896, 1865305376, 29782666544, 475549098160, 7593154541264, 121241257906000, 1935879286697296, 30910512661708432, 493553365105565264, 7880649886335326608, 125831666350680625104
Offset: 0

Views

Author

Gerhard Kirchner, Feb 12 2023

Keywords

Comments

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 12.

Links

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360575, A360577.

Formula

G.f.: (1-9*x+4*x^2-16*x^3) / (1-15*x-28*x^2+214*x^3-192*x^4-384*x^5+128*x^6).
Recurrence 1:
a(n) = 8*a(n-1) + 3*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 7*a(n-2)
b(n) = 12*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1) + e(n-1)
c(n) = 16*a(n-1) + 4*b(n-1) + 2*c(n-1)
d(n) = 2*a(n-1) + b(n-1) + d(n-1)
e(n) = 12*a(n-1) + 3*b(n-1)
with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
Recurrence 2:
a(n)=15*a(n-1) + 28*a(n-2) - 214*a(n-3) + 192*a(n-4) + 384*a(n-5) - 128*a(n-6)
for n>=6. For n<6, recurrence 1 can be used.

A360577 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 3, 60, 657, 8311, 101284, 1246049, 15292819, 187803572, 2305968393, 28315208039, 347681742812, 4269186204201, 52421329940803, 643681521419708, 7903765218510353, 97050331862075975, 1191681006432895780, 14632650860374551265, 179674317212728197891, 2206220907971874345652
Offset: 0

Views

Author

Gerhard Kirchner, Feb 12 2023

Keywords

Comments

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 13.

Links

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360575, A360576.

Formula

G.f.: (1-5*x-15*x^2-3*x^3+10*x^4) / (1-8*x-51*x^2-27*x^3+96*x^4+43*x^5-66*x^6).
Recurrence 1:
a(n) = 3*a(n-1) + b(n-1) + c(n-1) + 19*a(n-2) + 4*b(n-2) + c(n-2) + 2*d(n-2)
b(n) = 12*a(n-1) + 2*b(n-1) + 2*c(n-1) + e(n-1)
c(n) = 20*a(n-1) + 6*b(n-1) + 2*c(n-1) + 2*e(n-1)
d(n) = 4*a(n-1) + 2*b(n-1) + d(n-1)
e(n) = 24*a(n-1) + 7*b(n-1) + 2*c(n-1) + 2*d(n-1)
with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
Recurrence 2:
a(n)=8*a(n-1) + 51*a(n-2) + 27*a(n-3) - 96*a(n-4) - 43*a(n-5) + 66*a(n-6)
for n>=6. For n<6, recurrence 1 can be used.

A360644 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 1 X 1 dominos, 2 X 2 X 1 plates and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 12, 513, 16194, 547543, 18234354, 609298887, 20344385080, 679408772089, 22688284005780, 757662377924917, 25301659203704234, 844933359518672599, 28216027727373068302, 942256839186226313727, 31466085716246304261600, 1050790517091131646143477
Offset: 0

Views

Author

Gerhard Kirchner, Feb 15 2023

Keywords

Comments

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 14.

Links

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065.

Formula

G.f.: (1-16*x-18*x^2-13*x^3+10*x^4) / (1-28*x-195*x^2+497*x^3-30*x^4+79*x^5-66*x^6)
Recurrence 1:
a(n) = 12*a(n-1) + 4*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 43*a(n-2) + 8*b(n-2) + c(n-2) + 2*d(n-2)
b(n) = 32*a(n-1) + 9*b(n-1) + 4*c(n-1) + 2*d(n-1) + e(n-1)
c(n) = 60*a(n-1) + 16*b(n-1) + 6*c(n-1) + 4*d(n-1) + 2*e(n-1)
d(n) = 14*a(n-1) + 3*b(n-1) + d(n-1)
e(n) = 64*a(n-1) + 13*b(n-1) + 2*c(n-1) + 2*d(n-1)
with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
Recurrence 2:
a(n)=28*a(n-1) + 195*a(n-2) - 497*a(n-3) + 30*a(n-4) - 79*a(n-5) + 66*a(n-6)
for n>=6. For n<6, recurrence 1 can be used.
Showing 1-9 of 9 results.

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