A354131 -id:A354131

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A354130

Triangle read by rows: T(k,n) (k >= 0, n = 0, ..., k) = number of tilings of a k X n rectangle using 2 X 2, and 1 X 1 tiles, right trominoes and dominoes.

+10
2

1, 1, 1, 1, 2, 12, 1, 3, 48, 405, 1, 5, 216, 4185, 103300, 1, 8, 936, 40320, 2352830, 124098498, 1, 13, 4104, 397755, 55004286, 6763987198, 863829618636, 1, 21, 17928, 3892293, 1274945897, 364713815832, 108969107997657, 32100965172272499

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Tiling algorithm, see A351322.

Reading the sequence {T(k,n)} for n>k, use T(n,k) instead of T(k,n).

T(1,n) = A000045(n+1), Fibonacci numbers.

T(2,n) = A354131(n), T(3,n) = A354132(n).

LINKS

Table of n, a(n) for n=0..35.

EXAMPLE

Triangle begins

k\n_0__1____2______3________4__________5____________6

0: 1

1: 1 1

2: 1 2 12

3: 1 3 48 405

4: 1 5 216 4185 103300

5: 1 8 936 40320 2352830 124098498

6: 1 13 4104 397755 55004286 6763987198 863829618636

PROG

(Maxima), see A352589.

CROSSREFS

Cf. A000045, A351322, A354131, A354132.

KEYWORD

nonn,tabl

AUTHOR

Gerhard Kirchner, May 18 2022

STATUS

approved

A354132

Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles, right trominoes and dominoes.

+10
2

1, 3, 48, 405, 4185, 40320, 397755, 3892293, 38193444, 374425263, 3671810235, 36003770640, 353046480345, 3461866214283, 33946152068808, 332866572321933, 3263999126947497, 32005882711563552, 313840950402409011, 3077438640586986141, 30176522977460549436

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Tiling algorithm see A351322.

LINKS

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (6,38,0,-68,24,-3).

FORMULA

G.f.: (1 - 3*x - 8*x^2 + 3*x^3 - x^4) / (1 - 6*x - 38*x^2 + 68*x^4 - 24*x^5 + 3*x^6).

a(n) = 6*a(n-1) + 38*a(n-2) - 68*a(n-4) + 24*a(n-5) - 3*a(n-6).