This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
A(6,4) = A(4,6) = 3: ._._._._._._. ._._._._._._. ._._._._._._. | | | | | | | | | | |___|___|___| | |___| |___| | | | | | | | | | | | |___|___|___| |_______|___| |___|_______| . Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ... 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, ... 1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, ... 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, ... 1, 0, 1, 1, 3, 2, 7, 7, 16, 19, 40, ... 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 2, ... 1, 0, 1, 0, 5, 0, 16, 0, 48, 0, 160, ... 1, 0, 0, 1, 0, 0, 19, 0, 0, 50, 17, ... 1, 0, 1, 0, 8, 1, 40, 2, 160, 17, 796, ... ...
b:= proc(n, l) option remember; local i, k, 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))
else for k do if l[k]=0 then break fi od; s:=0;
for i from k+1 to 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]$j=i+1..nops(l)])
od; s
fi
end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..14);
b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which [Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = 0; For[i = k+1, i <= Length[l] && l[[i]] == 0, i++, s = s + b[n, Join [l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]] ]]]; s]]; 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, 14}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)
For n=5, there are 2 dissections where the orbits under the symmetry group of the square, D4, have 2 elements. For n=4, the 2 dissections can be seen in A240120 and A240121.
For n=5, there are 26 dissections where the orbits under the symmetry group of the square, D4, have 4 elements. The 6 dissections for n=4 can be seen in A240123 and A240125.
Triangle begins: n\k| 0 1 2 3 4 5 6 7 8 ---+----------------------------------------- 0 | 1 1 | 1 1 2 | 1 1 1 3 | 1 1 2 4 4 | 1 1 3 6 16 5 | 1 1 4 12 37 140 6 | 1 1 6 24 105 454 1987 7 | 1 1 10 40 250 1566 9856 62266 8 | 1 1 15 80 726 5670 47394 406168 3899340 A 5 X 4 rectangle can be tiled by 12 unit squares and 2 squares of side 2 in the following ways: +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | +---+---+---+---+ + +---+---+ +---+ +---+ +---+---+ + | | | | | | | | | | | | | | | | +---+---+ + +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | + +---+---+ + +---+---+ + +---+---+ + +---+---+ | | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ . +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | +---+ +---+ +---+---+---+---+ +---+---+---+---+ + +---+---+ | | | | | | | | | | | | | | | | | +---+---+---+---+ + +---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | + +---+---+ +---+ +---+ + + + + +---+---+ | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ . +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | | +---+---+ + +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | +---+---+---+---+ +---+ +---+ +---+---+---+---+ +---+---+ + | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ + + + + +---+---+ | | | | | | | | | | | | | | | + +---+---+ +---+ +---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ . +---+---+---+---+ | | | | +---+ +---+ | | | | +---+---+---+---+ | | | | | +---+---+---+---+ | | | | +---+ +---+ | | | | +---+---+---+---+ The first six of these have no symmetries, so they account for 4 tilings each. The next six have either a mirror symmetry or a rotational symmetry and account for 2 tilings each. The last has full symmetry and accounts for 1 tiling. In total there are 6*4+6*2+1 = 37 tilings. This is the maximum for a 5 X 4 rectangle, so T(5,4) = 37.
The irregular triangle begins: n,k\u 0 1 2 3 4 5 6 7 8 9 10 11 12 ... 1,1 1 2,1 1 2,2 1 1 3,1 1 3,2 1 2 3,3 1 4 0 0 1 4,1 1 4,2 1 3 1 4,3 1 6 4 0 2 4,4 1 9 16 8 5 0 0 0 0 1 5,1 1 5,2 1 4 3 5,3 1 8 12 0 3 4 5,4 1 12 37 34 15 12 4 0 0 2 5,5 1 16 78 140 88 44 68 32 0 4 0 0 0 ... ... For n = 4, k = 3, there are 4 tilings that contain 2 isolated nodes, so T(4,3,2) = 4. A 2 X 2 square contains 1 isolated node. Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors. Then the 4 tilings are: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
For n=5, there are 4 dissections where the orbits under the symmetry group of the square, D4, have 1 element. For n=4, 3 dissections divide the square into uniform subsquares (of sizes 1, 2 and 4 respectively), and this is the 4th: --------- | | | | | --------- | | | | --- --- | | | | --------- | | | | | ---------
For n=5, there are 45 dissections where the orbits under the symmetry group of the square, D4, have 8 elements. For n=4, this is the only dissection: --------- | | | | | ----- | | | ----- | | | | | --------- | | | | | ---------
The triangle is: n\k 1 2 3 4 5 6 7 8 ... . 0 1 1 1 1 1 1 1 1 ... 1 1 1 1 1 1 1 1 ... 2 1 3 2 5 3 8 ... 3 1 2 2 3 3 ... 4 3 12 5 25 ... 5 6 7 11 ... 6 17 106 ... 7 44 ... ... T(3,5) = 2 because there are 2 different tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will only transform each tiling into itself. Group D2 operations are: . the identity operation . rotation by 180 degrees . reflection about a horizontal axis through the center . reflection about a vertical axis through the center The tilings are: ._________. ._________. |_|_|_|_|_| |_| |_| |_|_|_|_|_| |_| |_| |_|_|_|_|_| |_|_____|_|
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