A233427
Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 5, 0, 0, 5, 0, 1, 1, 0, 0, 56, 0, 56, 0, 0, 1, 1, 0, 0, 0, 501, 501, 0, 0, 0, 1, 1, 0, 0, 0, 0, 4006, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 27950, 27950, 0, 0, 0, 1, 1, 1, 0, 45, 0, 0, 214689, 0, 214689, 0, 0, 45, 0, 1
Offset: 0
A(5,2) = A(2,5) = 5:
._________. ._________. ._________. ._________. ._________.
|_________| | ._____| | | |_____. | | ._| | | |_. |
|_________| |_|_______| |_______|_| |___|_____| |_____|___|.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 1, 0, ...
1, 0, 0, 0, 0, 5, 0, ...
1, 0, 0, 0, 0, 56, 0, ...
1, 0, 0, 0, 0, 501, 0, ...
1, 1, 5, 56, 501, 4006, 27950, ...
1, 0, 0, 0, 0, 27950, 0, ...
1, 0, 0, 0, 0, 214689, 0, ...
1, 0, 0, 0, 0, 1696781, 0, ...
1, 0, 0, 0, 0, 13205354, 0, ...
1, 1, 45, 7670, 890989, 101698212, 7845888732, ...
...
Row sums of
A247702,
A247703,
A247704,
A247705,
A247706,
A247707,
A247708,
A247709,
A247710,
A247711,
A247712,
A247713 give A(n,5).
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 *)
A247218
Number of tilings of a 15 X n rectangle using 3n pentominoes of shape I.
Original entry on oeis.org
1, 1, 1, 1, 1, 34, 95, 190, 325, 506, 3324, 10353, 25607, 55346, 108756, 389216, 1208901, 3281686, 8006108, 17950204, 51430928, 150609259, 419540401, 1090827453, 2651884943, 7077981621, 19691707908, 54499735145, 145671654672, 371632691473, 976543067070
Offset: 0
Showing 1-3 of 3 results.