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 *)
A233507
Number of tilings of a 2 X 4 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.
Original entry on oeis.org
1, 7, 201, 9787, 379688, 16512483, 726964790, 31549810845, 1378740599284, 60239603421159, 2630166605483293, 114886450998314920, 5017916294582867990, 219163121582772423673, 9572435654283943792842, 418094220600909382190818, 18261053013117932038592765
Offset: 0
a(1) = A219866(4,2) = A129682(4) = A219866(2,4) = A219862(2) = 7:
._______. ._______. ._______. ._______.
|_____| | | |_____| | | | | | |___| | |
|_____|_| |_|_____| |_|_|_|_| |___|_|_|
._______. ._______. ._______.
| |___| | | | |___| |___|___|
|_|___|_| |_|_|___| |___|___|.
-
b:= proc(n, l) option remember; local k, t; t:= min(l[]);
if n=0 then 1
elif t>0 then b(n-t, map(h->h-t, l))
else for k while l[k]>0 do od;
add(`if`(n>=j, b(n, s(k=j, l)), 0), j=2..3)+
`if`(k<=6 and l[k+2]=0, b(n, s(k=1, k+2=1, l)), 0)+
`if`(k<=4 and l[k+2]=0 and l[k+2*2]=0, b(n, s(k=1,
k+2=1, k+2*2=1, l)), 0)+ `if`(irem(k, 2)>0 and
l[k+1]=0, b(n, s(k=1, k+1=1, l)), 0)
fi
end:
a:=n-> b(n, [0$8]): s:= subsop:
seq(a(n), n=0..10);
-
b[n_, l_] := b[n, l] = Module[{k, t}, t = Min[l]; Which[n == 0, 1, t > 0, b[n-t, l-t], True, For[k = 1, l[[k]] > 0, k++]; Sum[If[n >= j, b[n, ReplacePart[l, k -> j]], 0], {j, 2, 3}] + If[k <= 6 && l[[k + 2]] == 0, b[n, ReplacePart[l, {k -> 1, k+2 -> 1}]], 0] + If[k <= 4 && l[[k+2]] == 0 && l[[k+2*2]] == 0, b[n, ReplacePart[l, {k -> 1, k+2 -> 1, k+2*2 -> 1}]], 0] + If[Mod[k, 2] > 0 && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 0]]]; a[n_] := b[n, Array[0&, 8]]; Table[Print[an = a[n]]; an, {n, 0, 16}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)
A202536
Number of tilings of a 4 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 0, 1, 3, 3, 8, 21, 31, 70, 165, 286, 615, 1351, 2548, 5353, 11343, 22320, 46349, 96516, 193944, 400313, 826747, 1678540, 3453642, 7105102, 14498569, 29781633, 61158957, 125108639, 256763850, 526846289, 1079030715, 2213527089, 4540131569, 9304062828
Offset: 0
a(3) = 3, because there are 3 tilings of a 4 X 3 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
._____. ._____. ._._._.
| | | | |_____| |_____|
| | | | | | | | |_____|
|_|_|_| | | | | |_____|
|_____| |_|_|_| |_____|
a(4) = 3, because there are 3 tilings of a 4 X 4 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
._._____. ._____._. ._._._._.
| |_____| |_____| | | . | . |
| | . | | | | . | | |___|___|
|_|___| | | |___|_| | . | . |
|_____|_| |_|_____| |___|___|
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1, 1, 11, -9, -8, -48, 29, 23, 115, -44, -31, -173, 35, 19, 174, -18, 2, -119, 10, -14, 56, -8, 12, -19, 4, -5, 5, -1, 1, -1).
-
gf:= -(x^3+x-1) *(x^18 -3*x^15 +x^14 +7*x^12 -3*x^11 -11*x^9 +3*x^8 +12*x^6 -x^5 -6*x^3+1) *(x-1)^2 *(x^2+x+1)^2 / (x^30 -x^29 +x^28 -5*x^27 +5*x^26 -4*x^25 +19*x^24 -12*x^23 +8*x^22 -56*x^21 +14*x^20 -10*x^19 +119*x^18 -2*x^17 +18*x^16 -174*x^15 -19*x^14 -35*x^13 +173*x^12 +31*x^11 +44*x^10 -115*x^9 -23*x^8 -29*x^7 +48*x^6 +8*x^5 +9*x^4 -11*x^3 -x^2 -x+1):
a:= n-> coeff(series(gf, x, n+1),x,n);
seq(a(n), n=0..50);
Showing 1-3 of 3 results.