A219866
Number A(n,k) of tilings of a k X n rectangle using dominoes and straight (3 X 1) trominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 4, 1, 1, 1, 2, 7, 14, 7, 2, 1, 1, 2, 15, 41, 41, 15, 2, 1, 1, 3, 30, 143, 184, 143, 30, 3, 1, 1, 4, 60, 472, 1069, 1069, 472, 60, 4, 1, 1, 5, 123, 1562, 5624, 9612, 5624, 1562, 123, 5, 1
Offset: 0
A(2,3) = A(3,2) = 4, because there are 4 tilings of a 3 X 2 rectangle using dominoes and straight (3 X 1) trominoes:
.___. .___. .___. .___.
| | | |___| | | | |___|
| | | |___| |_|_| | | |
|_|_| |___| |___| |_|_|
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 1, 1, 1, 2, 2, 3, ...
1, 1, 2, 4, 7, 15, 30, 60, ...
1, 1, 4, 14, 41, 143, 472, 1562, ...
1, 1, 7, 41, 184, 1069, 5624, 29907, ...
1, 2, 15, 143, 1069, 9612, 82634, 707903, ...
1, 2, 30, 472, 5624, 82634, 1143834, 15859323, ...
1, 3, 60, 1562, 29907, 707903, 15859323, 354859954, ...
Columns (or rows) k=0-10 give:
A000012,
A000931(n+3),
A129682,
A219867,
A219862,
A219868,
A219869,
A219870,
A219871,
A219872,
A219873.
-
b:= proc(n, l) option remember; local k, t;
if max(l[])>n then 0 elif n=0 or l=[] then 1
elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
else for k do if l[k]=0 then break fi od;
b(n, subsop(k=3, l))+ b(n, subsop(k=2, l))+
`if`(k `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..10);
-
b[n_, l_] := b[n, l] = Module[{k, t}, If [Max[l] > n, 0, If[ n == 0 || l == {}, 1, If[Min[l] > 0, t = Min[l]; b[n-t, l-t], k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 3]] + b[n, ReplacePart[l, k -> 2]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1, k+2 -> 1}]], 0]]]]]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)
A233807
Number of tilings of an n X n square using right trominoes and at most one monomino.
Original entry on oeis.org
1, 1, 4, 0, 16, 128, 162, 34528, 943096, 1193600, 3525377600, 480585761344, 2033502499954, 46983507796973152, 32908187880881958736, 458324092996867592192, 83153202122213272708832688, 299769486068040749617049301344
Offset: 0
a(2) = 4:
.___. .___. .___. .___.
|_| | | |_| | ._| |_. |
|___| |___| |_|_| |_|_| .
-
b:= proc(n, w, l) option remember; local k, t;
if max(l[])>n then 0 elif n=0 then 1
elif min(l[])>0 then t:=min(l[]); b(n-t, w, map(h->h-t, l))
else for k while l[k]>0 do od;
`if`(w, b(n, false, s(k=1, l)), 0)+
`if`(k>1 and l[k-1]=1, b(n, w, s(k=2, k-1=2, l)), 0)+
`if`(k b(n, evalb(irem(n, 3)>0), [0$n]): s:= subsop:
seq(a(n), n=0..10);
-
$RecursionLimit = 1000; s = ReplacePart; b[n_, w_, l_] := b[n, w, l] = Module[{k, t}, Which[Max[l] > n, 0,n == 0, 1,Min[l] > 0, t = Min[l]; b[n-t, w, l-t], True, For[k = 1, l[[k]] > 0, k++ ]; If[w, b[n, False, s[l, k -> 1]], 0]+If[k > 1 && l[[k-1]] == 1, b[n, w, s[l, {k -> 2, k-1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 1, b[n, w, s[l, {k -> 2, k+1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 0, b[n, w, s[l, {k -> 1, k+1 -> 2}]]+b[n, w, s[l, {k -> 2, k+1 -> 1}]] + If[w, b[n, False, s[l, {k -> 2, k+1 -> 2}]], 0], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, w, s[l, {k -> 2, k+1 -> 2, k+2 -> 2}]], 0] ] ]; a[n_] := b[n, Mod[n, 3] > 0, Array[0 &, n]]; Table[Print[an = a[n]]; an, {n, 0, 16}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)
A364504
Number of tilings of an n X n square using dominoes and trominoes (of any shape).
Original entry on oeis.org
1, 0, 2, 30, 1352, 226922, 128441094, 267855152858, 1990875917805852, 52918918728713551244, 5046393600526600826576990, 1722423352379200292770200349728, 2106175531602971710801901685263906946, 9224774497259881798661234516031754156588512, 144713995908210595144464778124853904625705818728754
Offset: 0
A353777
Number of tilings of an n X n square using dominoes, monominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 1, 8, 163, 15623, 5684228, 8459468955, 50280716999785, 1202536689448371122, 115462301811597894998929, 44537596159273736617786474211, 69003082378039459280864860681919942, 429429579883061866326542598342441907826951, 10734684843612889640707750537898705644071715970757
Offset: 0
a(2) = 8:
.___. .___. .___. .___. .___. .___. .___. .___.
| | |_|_| |___| | | | |_|_| |___| |_| | | |_|
|___| |_|_| |___| |_|_| |___| |_|_| |_|_| |_|_| .
A353934
Number of tilings of an n X n square using right trominoes, dominoes, and monominoes.
Original entry on oeis.org
1, 1, 11, 369, 83374, 90916452, 546063639624, 17259079054003609, 2916019543694306398589, 2620143594924539083433405392, 12541344781693990981151732534871036, 319608708168951734031266758322647453517098, 43373075269161087186367095378869660507262626652634
Offset: 0
a(2) = 11:
.___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
|_|_| |___| | | | |_|_| |___| |_| | | |_| |_| | |_. | | ._| | |_|
|_|_| |___| |_|_| |___| |_|_| |_|_| |_|_| |___| |_|_| |_|_| |___| .
Cf.
A004003,
A028420,
A063443,
A219874,
A219952,
A219994,
A220061,
A220778,
A233807,
A270071,
A353777.
Showing 1-5 of 5 results.