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Search: a220778 -id:a220778
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A220777
Number A(n,k) of tilings of a k X n rectangle using integer-sided rectangular tiles of equal area; square array A(n,k), n>=0, k>=0, read by antidiagonals.
+10
11
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 2, 9, 4, 9, 2, 1, 1, 4, 11, 20, 20, 11, 4, 1, 1, 2, 21, 7, 49, 7, 21, 2, 1, 1, 4, 24, 54, 115, 115, 54, 24, 4, 1, 1, 3, 43, 12, 343, 4, 343, 12, 43, 3, 1, 1, 4, 62, 190, 850, 1225, 1225, 850, 190, 62, 4, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Alois P. Heinz, Antidiagonals n = 0..26, flattened
EXAMPLE
A(3,5) = 7, because there are 7 tilings of a 5 X 3 rectangle using integer-sided rectangular tiles of equal area:
._____. ._____. ._____. ._____. ._____. ._____. ._____.
| | | | | | |_____| |_____| |_____| | | | | |_|_|_|
| | | | | | |_____| |_____| | | | | | | | | |_|_|_|
| | | | | | |_____| | | | | | | | | |_|_|_| |_|_|_|
| | | | | | |_____| | | | | |_|_|_| |_____| |_|_|_|
|_____| |_|_|_| |_____| |_|_|_| |_____| |_____| |_|_|_|
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 2, 3, 2, 4, 2, 4, ...
1, 2, 4, 6, 9, 11, 21, 24, 43, ...
1, 2, 6, 4, 20, 7, 54, 12, 190, ...
1, 3, 9, 20, 49, 115, 343, 850, 2401, ...
1, 2, 11, 7, 115, 4, 1225, 7, 15242, ...
1, 4, 21, 54, 343, 1225, 7104, 31777, 169952, ...
1, 2, 24, 12, 850, 7, 31777, 4, 1300180, ...
1, 4, 43, 190, 2401, 15242, 169952, 1300180, 13036591, ...
...
MAPLE
b:= proc(n, l, d) option remember; local i, k, m, q, s, 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), d)
else for k do if l[k]=0 then break fi od; s, m:=0, nops(l);
for i from k to m while l[i]=0 do if irem(d, 1+i-k, 'q')=0
and q<=n then s:= s+ b(n, [l[j]$j=1..k-1, q$j=k..i,
l[j]$j=i+1..m], d) fi od; s
fi
end:
A:= (n, k)-> `if`(n<k, A(k, n), `if`(k=0, 1,
add(b(n, [0$k], d), d=numtheory[divisors](n*k)))):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
$RecursionLimit = 1000; b[n_, l_, d_] := b[n, l, d] = Module[{i, k, m, q, s, t}, Which[ Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t, d], True, k = Position[l, 0, 1][[1, 1]]; {s, m} = {0, Length[l]}; For[i = k, i <= m && l[[i]] == 0, i++, If[(Mod[d, 1+i-k]) == 0 && (q = Quotient[d, 1+i-k]) <= n, s = s + b[n, Join[l[[1 ;; k-1]], Table[q, {j, k, i}], l[[i+1 ;; m]]], d] ] ]; s ] ]; a[n_, k_] := a[n, k] = If[n < k, a[k, n], If[k == 0, 1, Sum[b[n, Array[0&, k], d], {d, Divisors[n*k]}]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)
CROSSREFS
Columns (or rows) k=0-10 give: A000012, A000005, A220768, A220769, A220770, A220771, A220772, A220773, A220774, A220775, A220776.
Main diagonal gives: A220778.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Dec 19 2012
STATUS
approved
A353777
Number of tilings of an n X n square using dominoes, monominoes and 2 X 2 tiles.
+10
3
1, 1, 8, 163, 15623, 5684228, 8459468955, 50280716999785, 1202536689448371122, 115462301811597894998929, 44537596159273736617786474211, 69003082378039459280864860681919942, 429429579883061866326542598342441907826951, 10734684843612889640707750537898705644071715970757
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..17
Wikipedia, Polyomino
FORMULA
a(n) = A352589(n,n).
EXAMPLE
a(2) = 8:
.___. .___. .___. .___. .___. .___. .___. .___.
| | |_|_| |___| | | | |_|_| |___| |_| | | |_|
|___| |_|_| |___| |_|_| |___| |_|_| |_|_| |_|_| .
CROSSREFS
Main diagonal of A352589.
Cf. A004003, A028420, A063443, A219874, A219952, A219994, A220061, A220778, A233807, A270071.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 07 2022
STATUS
approved
A353934
Number of tilings of an n X n square using right trominoes, dominoes, and monominoes.
+10
2
1, 1, 11, 369, 83374, 90916452, 546063639624, 17259079054003609, 2916019543694306398589, 2620143594924539083433405392, 12541344781693990981151732534871036, 319608708168951734031266758322647453517098, 43373075269161087186367095378869660507262626652634
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..16
Wikipedia, Polyomino
FORMULA
a(n) = A353877(n,n).
EXAMPLE
a(2) = 11:
.___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
|_|_| |___| | | | |_|_| |___| |_| | | |_| |_| | |_. | | ._| | |_|
|_|_| |___| |_|_| |___| |_|_| |_|_| |_|_| |___| |_|_| |_|_| |___| .
CROSSREFS
Main diagonal of A353877.
Cf. A004003, A028420, A063443, A219874, A219952, A219994, A220061, A220778, A233807, A270071, A353777.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 11 2022
STATUS
approved
A363381
a(n) is the number of distinct n-cell patterns that tile an n X n square.
+10
1
1, 2, 1, 60, 1, 102, 1, 62714
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Consider n unit squares contained within an n X n square. The n unit squares are an n-cell pattern of the n X n square if, when copied n-1 times, they can, with various rigid transformations, be combined to tessellate the n X n square.
Put another way:
Consider, for example, for n = 4, a transparency with an outline of a 4 X 4 square filled by 16 unit squares. Any 4 unit squares are painted the same color. Those four squares are a potential n-cell pattern of the 4 X 4 square. Three copies of the transparency are made with only the color of the 4 squares being different. If a person can stack the transparencies in such a way that they fill the 4 X 4 square, then the n-cell pattern is acceptable.
Put another way:
n unit squares from an n X n square are painted a color. Those n unit squares are an n-cell pattern. If n-1 copies of the pattern can be painted (each a different color) and together they fill the n X n square, then the n unit squares form an acceptable n-cell pattern.
Conjecture by Andrew Young: For an n X n square, where n is an odd prime, there is only one n-cell pattern.
Conjecture by Andrew Young and Thomas Young: An odd integer n>=3 is prime iff there exists only one n-cell pattern for an n X n square.
For composite numbers n, an n X n square will always have at least two n-cell patterns: a 1 X n pattern and an f1 X f2 pattern, where 1 < f1 <= f2 < n and f1*f2 = n. For example, a 14 X 14 square can be tiled using fourteen 1 X 14 rectangles or fourteen 2 X 7 rectangles; a 15 X 15 square can be tiled using fifteen 1 X 15 rectangles or fifteen 3 X 5 rectangles; a 9 X 9 square can be tiled using nine 1 X 9 rectangles or nine 3 X 3 squares (as in Sudoku!).
For prime numbers p, a p X p square can always be tessellated with p rectangles that are 1 X p.
LINKS
Table of n, a(n) for n=1..8.
Thomas Young, The 60 4-cell patterns for a 4 X 4 square.
Thomas Young, A Java program to calculate the number of 4-cell pattern for a 4 X 4 square.
Thomas Young, The 102 6-cell patterns for a 6 X 6 square.
EXAMPLE
For n = 1, there is one 1-cell pattern because there is only one unit square to paint.
For n = 2, there are two 2-cell patterns:
+---+---+ +---+---+ +---+
| 1 | 2 | | 1 | 2 | | 1 |
+---+---+ +---+---+ and +---+---+
| 3 | 4 | | 4 |
+---+---+ +---+
For n = 3, there is one 3-cell pattern:
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 | it is +---+---+---+
+---+---+---+ | 1 | 2 | 3 |
| 7 | 8 | 9 | +---+---+---+
+---+---+---+
For n = 4, there are sixty 4-cell patterns:
+---+---+---+---+
| 1 | 2 | 3 | 4 |
+---+---+---+---+
| 5 | 6 | 7 | 8 | one is +---+---+---+---+
+---+---+---+---+ | 1 | 2 | 3 | 4 |
| 9 |10 |11 |12 | +---+---+---+---+
+---+---+---+---+
|13 |14 |15 |16 |
+---+---+---+---+
+---+---+---+---+ +---+
| 1 | 2 | 3 | 4 | is equivalent to | 1 |
+---+---+---+---+ +---+
| 5 |
+---+
| 9 |
+---+
|13 |
+---+
and therefore is counted as one pattern.
Another 4-cell pattern for a 4 X 4
+---+---+---+---+
| x | x | y | y |
+---+---+---+---+
| z | y | x | a | is +---+---+
+---+---+---+---+ | x | x |
| y | z | a | x | +---+---+---+
+---+---+---+---+ | x |
| a | a | z | z | +---+---+
+---+---+---+---+ | x |
+---+
+---+---+
| x | x |
+---+---+---+ is equivalent to
| x |
+---+---+
| x |
+---+
+---+---+ +---+ +---+
| y | y | | z | | a |
+---+---+---+ +---+---+ +---+---+
| y | | z | | a |
+---+---+ +---+---+---+ +---+---+---+
| y | | z | z | | a | a |
+---+ +---+---+ +---+---+
because the shapes can be created through reflection, rotation, or translation.
Therefore, they are counted as one pattern.
For n = 5, there is one 5-cell pattern.
CROSSREFS
Cf. A291806, A227004, A291808, A360630, A291807, A328020, A220778.
KEYWORD
nonn,more,hard
AUTHOR
Thomas Young, May 30 2023
EXTENSIONS
a(7)-a(8) from Andrew Howroyd, Jun 04 2023
STATUS
approved
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