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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.

User: Oluwatobi Jemima Alabi

Oluwatobi Jemima Alabi's wiki page.

Oluwatobi Jemima Alabi has authored 3 sequences.

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

Views

Author

Greg Dresden and Oluwatobi Jemima Alabi, Jul 06 2020

Keywords

Comments

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).

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    [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
  • Mathematica
    a[n_]:= (2/3)*(Fibonacci[n-1, 2] - (-1)^n*Fibonacci[n-1]);
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 15 2022 *)
  • PARI
    concat([0,0] , Vec(2*x^3/((1+x-x^2)*(1-2*x-x^2)) + O(x^30))) \\ Colin Barker, Aug 06 2020
  • Sage
    [(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

Formula

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

A335747 Number of ways to tile vertically-fault-free 3 X n strip with squares and dominoes.

Original entry on oeis.org

1, 3, 13, 26, 66, 154, 380, 904, 2204, 5286, 12818, 30854, 74636, 179948, 434820, 1049122, 2533818, 6115538, 14766868, 35646080, 86064196, 207766110, 501609946, 1210964110, 2923573588, 7058053972, 17039774268
Offset: 0

Views

Author

Greg Dresden and Oluwatobi Jemima Alabi, Jun 20 2020

Keywords

Comments

By "vertically-fault-free" we mean that the tilings of the 3 X n strip do not split along any interior vertical line. Here are two of the 66 possible vertically-fault-free tilings of a 3 X 4 strip with squares and dominoes:
. _ _
| |_ | | _|_| |
|| _| | | | |_|
|| _|| ||_| |

Examples

			a(2) = 13 thanks to these thirteen vertically-fault-free tilings of a 3 X 2 strip:
._ _     _ _     _ _     _ _     _ _     _ _     _ _
|_ _|   |_|_|   |_|_|   |_ _|   |_|_|   |_ _|   |_ _|
|_|_|   |_ _|   |_|_|   |_ _|   |_ _|   |_|_|   |_ _|
|_|_|   |_|_|   |_ _|   |_|_|   |_ _|   |_ _|   |_ _|
._ _     _ _     _ _     _ _     _ _     _ _
|_ _|   |_ _|   |_ _|   | |_|   |_| |   | | |
| |_|   |_| |   | | |   |_|_|   |_|_|   |_|_|
|_|_|   |_|_|   |_|_|   |_ _|   |_ _|   |_ _|

Links

Crossrefs

Cf. A033506 (which gives all tilings of 3 X n strip), A112577, A134438, A291227.
Cf. A000045, A000129.

Programs

  • Magma
    I:=[26, 66, 154, 380]; [1,3,13] cat [n le 4 select I[n] else Self(n-1) +4*Self(n-2) -Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Jan 15 2022
  • Mathematica
    CoefficientList[Series[(1+2x+6x^2+2x^3-8x^4+x^6)/((1+x-x^2)(1-2x-x^2)), {x, 0, 26}], x] (* Michael De Vlieger, Jul 03 2020 *)
LinearRecurrence[{1,4,-1,-1}, {1,3,13,26,66,154,380}, 40] (* G. C. Greubel, Jan 15 2022 *)
  • Sage
    def P(n): return lucas_number1(n,2,-1)
def A335747(n): return (1/3)*(-9*bool(n==0) - 3*bool(n==1) + 3*bool(n==2) + 2*(3*P(n+1) + 2*P(n-1)) + 2*(-1)^n*fibonacci(n-1))
[A335747(n) for n in (0..40)] # G. C. Greubel, Jan 15 2022

Formula

a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4) for n >= 7.
a(n) = 2*A291227(n) - 8*A112577(n-2) + 2*A112577(n-4) for n >= 4.
a(n) = (2/3)*(A221174(n+1) + (-1)^n*A000045(n-1)) for n >= 3. - Greg Dresden, Jul 03 2020
G.f.: (1 + 2*x + 6*x^2 + 2*x^3 - 8*x^4 + x^6) / ((1 + x - x^2)*(1 - 2*x - x^2)). - Colin Barker, Jun 21 2020
a(n) = (1/3)*(-9*[n=0] - 3*[n=1] + 3*[n=2] + 2*(3*A000129(n+1) + 2*A000129(n-1)) + 2*(-1)^n*Fibonacci(n-1)). - G. C. Greubel, Jan 15 2022

A335560 Number of ways to tile an n X n square with 1 X 1 squares and (n-1) X 1 vertical or horizontal strips.

Original entry on oeis.org

1, 16, 131, 335, 851, 2207, 5891, 16175, 45491, 130367, 378851, 1112015, 3286931, 9762527, 29091011, 86879855, 259853171, 777986687, 2330814371, 6986151695, 20945872211, 62812450847, 188387020931, 565060399535, 1694979872051, 5084536963007, 15252805582691
Offset: 1

Views

Author

Oluwatobi Jemima Alabi, Jun 14 2020

Keywords

Comments

It is assumed that 1 X 1 squares and 1 X 1 strips can be distinguished. - Alois P. Heinz, Feb 23 2022

Examples

			Here is one of the 131 ways to tile a 3 X 3 square, in this case using two horizontal and two vertical strips:
   _ _ _
  |_ _| |
  | |_|_|
  |_|_ _|

Links

Crossrefs

Cf. A063443 and A211348 (tiling an n X n square with smaller squares).
Cf. A028420 (tiling an n X n square with monomers and dimers).

Programs

  • Mathematica
    Join[{1, 16}, LinearRecurrence[{6, -11, 6}, {131, 335, 851}, 25]] (* Amiram Eldar, Jun 16 2020 *)
  • PARI
    Vec(x*(1 + 10*x + 46*x^2 - 281*x^3 + 186*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Jun 14 2020

Formula

a(n) = 2*3^n + 12*2^n - 19, for n >= 3.
From Colin Barker, Jun 14 2020: (Start)
G.f.: x*(1 + 10*x + 46*x^2 - 281*x^3 + 186*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>5. (End)
E.g.f.: 5 - 19*exp(x) +12 *exp(2*x) + 2*exp(3*x) - 10*x - 31*x^2/2. - Stefano Spezia, Aug 25 2025

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