A250662
Number A(n,k) of tilings of a 2k X n rectangle using 2n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 6, 36, 1, 1, 1, 1, 1, 1, 13, 95, 1, 1, 1, 1, 1, 1, 7, 22, 281, 1, 1, 1, 1, 1, 1, 1, 15, 64, 781, 1, 1, 1, 1, 1, 1, 1, 8, 25, 155, 2245, 1, 1, 1, 1, 1, 1, 1, 1, 17, 37, 321, 6336, 1, 1
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 5, 1, 1, 1, 1, 1, 1, ...
1, 1, 11, 6, 1, 1, 1, 1, 1, ...
1, 1, 36, 13, 7, 1, 1, 1, 1, ...
1, 1, 95, 22, 15, 8, 1, 1, 1, ...
1, 1, 281, 64, 25, 17, 9, 1, 1, ...
1, 1, 781, 155, 37, 28, 19, 10, 1, ...
1, 1, 2245, 321, 100, 41, 31, 21, 11, ...
Columns k=0+1,2-10 give:
A000012,
A005178(n+1),
A236577,
A236582,
A247117,
A250663,
A250664,
A250665,
A250666,
A250667.
-
b:= proc(n, l) option remember; local d, k; d:= nops(l)/2;
if n=0 then 1
elif min(l[])>0 then (m->b(n-m, map(x->x-m, l)))(min(l[]))
else for k while l[k]>0 do od;
`if`(nd+1 or max(l[k..k+d-1][])>0, 0,
b(n, [l[1..k-1][],1$d,l[k+d..2*d][]]))
fi
end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$2*k])):
seq(seq(A(n,d-n), n=0..d), d=0..14);
-
b[n_, l_List] := b[n, l] = Module[{d = Length[l]/2, k}, Which[n == 0, 1, Min[l] > 0 , Function[{m}, b[n-m, l-m]][Min[l]], True, For[k=1, l[[k]] > 0, k++]; If[n d]]] + If[d == 1 || k > d+1 || Max[l[[k ;; k+d-1]]] > 0, 0, b[n, Join[l[[1 ;; k-1]], Array[1&, d], l[[k+d ;; 2*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 2k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)
A236576
The number of tilings of a 5 X (3n) floor with 1 X 3 trominoes.
Original entry on oeis.org
1, 4, 22, 121, 664, 3643, 19987, 109657, 601624, 3300760, 18109345, 99355414, 545105209, 2990674357, 16408085929, 90021597712, 493896002842, 2709719309845, 14866649448256, 81564634762843, 447497579542135
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.
- R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 20.
- R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014.
- Index entries for linear recurrences with constant coefficients, signature (6,-3,1).
-
g := (1-x)^2/(1-6*x+3*x^2-x^3) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;
-
CoefficientList[Series[(1 - x)^2/(1 - 6 x + 3 x^2 - x^3), {x,0,50}], x] (* G. C. Greubel, Apr 29 2017 *)
LinearRecurrence[{6, -3, 1}, {1, 4, 22}, 30] (* M. Poyraz Torcuk, Nov 06 2021 *)
-
my(x='x+O('x^50)); Vec((1-x)^2/(1-6*x+3*x^2-x^3)) \\ G. C. Greubel, Apr 29 2017
A236578
The number of tilings of a 7 X (3n) floor with 1 X 3 trominoes.
Original entry on oeis.org
1, 9, 155, 2861, 52817, 972557, 17892281, 329097125, 6052932495, 111328274273, 2047599783121, 37660384283749, 692666924307063, 12739845501187821, 234317040993180833, 4309665744385061493, 79265335342431559977
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..785
- R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 22.
- R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq. (15).
-
p := (x-1)^2*(-x^15 +14*x^14 -104*x^13 +527*x^12 -1971*x^11 +5573*x^10 -11973*x^9 +19465*x^8 -23695*x^7 +21166*x^6 -13512*x^5 +5915*x^4 -1685*x^3 +291*x^2 -27*x+1) ;
q := -17*x^17 +293180*x^8 -236178*x^7 +142400*x^6 -62621*x^5 +19420*x^4 -4062*x^3 +533*x^2 -38*x +x^18 +1 +151*x^16 -946*x^15 +4558*x^14 -17135*x^13 +50164*x^12 -114198*x^11 +202080*x^10 -277277*x^9 ;
taylor(p/q,x=0,30) ;
gfun[seriestolist](%) ;
A251073
Number of tilings of a 9 X n rectangle using 3n trominoes of shape I.
Original entry on oeis.org
1, 1, 1, 19, 57, 121, 783, 2861, 8133, 37160, 143419, 468816, 1876855, 7263468, 25496863, 97187247, 372086645, 1352780401, 5071962134, 19220628318, 71025008365, 265095817718, 997839772024, 3713274525679, 13851695644227, 51940567251136, 193830054345968
Offset: 0
Showing 1-4 of 4 results.
Comments