A226936
Number T(n,k) of squares of size k^2 in all tilings of an n X n square using integer-sided square tiles; triangle T(n,k), n >= 1, 1 <= k <= n, read by rows.
Original entry on oeis.org
1, 4, 1, 29, 4, 1, 312, 69, 4, 1, 5598, 1184, 153, 4, 1, 176664, 40078, 4552, 373, 4, 1, 9966344, 2311632, 285414, 18160, 917, 4, 1, 1018924032, 241967774, 30278272, 2128226, 74368, 2321, 4, 1, 190191337356, 45914039784, 5860964300, 411308056, 16210982, 311784, 5933, 4, 1
Offset: 1
For n=3 there are [29, 4, 1] squares of sizes [1^2, 2^2, 3^3] in all tilings of a 3 X 3 square:
._._._. ._._._. ._._._. ._._._. ._._._. ._._._.
| | | |_| |_|_|_| |_| | |_|_|_| |_|_|_|
| | |___|_| | |_| |_|___| |_| | |_|_|_|
|_____| |_|_|_| |___|_| |_|_|_| |_|___| |_|_|_|.
Triangle T(n,k) begins:
n \ k 1 2 3 4 5 6 7 8
--:----------------------------------------------------------------
1 : 1;
2 : 4, 1;
3 : 29, 4, 1;
4 : 312, 69, 4, 1;
5 : 5598, 1184, 153, 4, 1;
6 : 176664, 40078, 4552, 373, 4, 1;
7 : 9966344, 2311632, 285414, 18160, 917, 4, 1;
8 : 1018924032, 241967774, 30278272, 2128226, 74368, 2321, 4, 1;
-
b:= proc(n, l) option remember; local i, k, s, t;
if max(l[])>n then [0$2] elif n=0 then [1, 0]
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; s:=[0$2];
for i from k to nops(l) while l[i]=0 do s:= s+(h->h+
[0, h[1]*x^(1+i-k)])(b(n, [l[j]$j=1..k-1,
1+i-k$j=k..i, l[j]$j=i+1..nops(l)])) od; s
fi
end:
T:= n-> seq(coeff(b(n, [0$n])[2],x,k), k=1..n):
seq(T(n), n=1..10);
-
$RecursionLimit = 1000; b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, {0, 0}, n == 0, {1, 0}, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1, 1][[1, 1]]; s = {0, 0}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s + Function[h, h + {0, h[[1]]*x^(1+i-k)}][b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]] ] ] ] ]; s] ]; T[n_] := Table[Coefficient[b[n, Array[0&, n]][[2]], x, k], {k, 1, n}]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Dec 23 2013, translated from Maple *)
A211348
Number of ways to tile an n X n square with 1 X 1, 2 X 2 and 3 X 3 tiles.
Original entry on oeis.org
1, 1, 2, 6, 39, 467, 10290, 431842, 33702357, 4933399675, 1353257600290, 694985665826606, 668743276018647665, 1205268925168096642391, 4069023157203412697840109, 25732126785058509461002703360, 304814553338563601845965453449729
Offset: 0
-
b:= proc(n, l) option remember; local i, k, s, t;
if max(l[])>n then 0 elif n=0 then 1
elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
else for k do if l[k]=0 then break fi od; s:=0;
for i from k to min(k+2, nops(l)) while l[i]=0 do s:=s+
b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
od; s
fi
end:
a:= n-> b(n, [0$n]):
seq(a(n), n=0..10); # Alois P. Heinz, Feb 05 2013
A240120
Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has reflective symmetry in both diagonals and no other reflective symmetries.
Original entry on oeis.org
0, 0, 0, 1, 1, 9, 19, 121, 275, 2489, 7217, 86775
Offset: 1
This is the single dissection for n=4:
---------
| | | |
| -----
| | | |
---------
| | | |
----- |
| | | |
---------
A240121
Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has two reflective symmetries in axes parallel to the sides, and no other reflective symmetries.
Original entry on oeis.org
0, 0, 0, 1, 0, 13, 5, 183, 75, 4408, 1501, 180324
Offset: 1
This dissection is the only example for n=4:
---------
| | | |
--- ---
| | | |
---------
| | | |
--- ---
| | | |
---------
A240122
Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 90-degree rotational symmetry and no reflective symmetry.
Original entry on oeis.org
0, 0, 0, 0, 1, 2, 12, 40, 154, 760, 3260, 22730
Offset: 1
The two dissections for n=6:
------------- -------------
| | | | | | | | | | | |
--- ------- --- -------
| | | | | | | | | |
--------- | --------- |
| | | | | | | | | | |
----- ----- -------------
| | | | | | | | | | |
| --------- ---------
| | | | | | | | | |
------- --- ------- ---
| | | | | | | | | | | |
------------- -------------
A240123
Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has a reflective symmetry in one diagonal, but no other symmetries.
Original entry on oeis.org
0, 0, 1, 3, 19, 107, 847, 8647, 119835, 2255123, 58125783, 2050662011
Offset: 1
The three dissections for n=4:
--------- --------- ---------
| | | | | | | | | |
| ----- | | | | ---
| | | | | | | | | |
--------- --------- | ---
| | | | | | | | | | | |
--------- | ----- ---------
| | | | | | | | | | | | | |
--------- --------- ---------
A240124
Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 180-degree rotational symmetry, but no other symmetries.
Original entry on oeis.org
0, 0, 0, 0, 2, 19, 109, 1781, 13660, 397689, 5368943, 289864745
Offset: 1
The two dissections for n=5:
----------- -----------
| | | | | | | | |
| | --- --- -----
| | | | | | | | |
----------- -----------
| | | | | | | | | | | |
----------- -----------
| | | | | | | | |
--- | | ----- ---
| | | | | | | | |
----------- -----------
A240125
Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has one reflective symmetry in an axis parallel to a side, but no other symmetries.
Original entry on oeis.org
0, 0, 0, 3, 5, 138, 201, 13032, 19990, 4095612, 7026883, 4451051502
Offset: 1
The three dissections for n=4, with the axis horizontal:
--------- --------- ---------
| | | | | | | | | | | | |
| ----- | ----- ---------
| | | | | | | | | | |
--------- ----- | | -----
| | | | | | | | | | |
| ----- | ----- ---------
| | | | | | | | | | | | |
--------- --------- ---------
A221845
Number of prime dissections of an n X n square into integer-sided squares.
Original entry on oeis.org
1, 1, 5, 38, 471, 10661, 450923, 35863932, 5353011030, 1500957421749, 790347882174803, 781621363452395224, 1451740730942350766747, 5064070747064013555843107, 33176273260130056822126522407
Offset: 1
For n = 3 the a(3) = 5 dissections are:
+-+-+-+ +-+-+-+ +-+-+-+ +-+---+ +---+-+
| | | | | | | | | | | | | | | | | |
+-+-+-+ +-+-+-+ +-+-+-+ +-+ | | +-+
| | | | | | | | | | | | | | | |
+-+-+-+ +-+ | | +-+ +-+-+-+ +-+-+-+
| | | | | | | | | | | | | | | | | |
+-+-+-+ +-+---+ +---+-+ +-+-+-+ +-+-+-+
- J. H. Conway, Mrs Perkins's quilt, Proc. Camb. Phil. Soc., 60 (1964), 363-368.
A226897
a(n) is the total number of parts in the set of partitions of an n X n square lattice into squares, considering only the list of parts.
Original entry on oeis.org
1, 5, 16, 59, 156, 529, 1351, 3988, 10236, 27746, 66763, 176783, 412450
Offset: 1
For n = 3, the partitions are:
Square side 1 2 3 Total Parts
9 0 0 9
5 1 0 6
0 0 1 1
Total 16
So a(3) = 16.
-
b:= proc(n, l) option remember; local i, k, s, t;
if max(l[])>n then {} elif n=0 or l=[] then {0}
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; s:={};
for i from k to nops(l) while l[i]=0 do s:=s union
map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,
1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
od; s
fi
end:
a:= n-> add(coeff(add(j, j=b(n, [0$n])), x, i), i=1..n):
seq(a(n), n=1..9); # Alois P. Heinz, Jun 21 2013
-
$RecursionLimit = 1000; b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which [Max[l]>n, {}, n == 0 || l == {}, {0}, Min[l]>0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1, 1][[1, 1]]; s = {}; For[i = k, i <= Length[l] && l[[i]]== 0, i++, s = s ~Union~ Map[Function[{v}, v+x^(1+i-k)], b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]] ]]]]; s]]; a[n_] := Sum[Coefficient[Sum[j, {j, b[n, Array[0&, n]]}], x, i], {i, 1, n}]; Table[a[n], {n, 1, 9}] (* Jean-François Alcover, May 29 2015, after Alois P. Heinz *)
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