A226206 Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles of area > 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 3, 1, 3, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 5, 0, 7, 0, 5, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 7, 7, 0, 0, 0, 0, 0, 1
Offset: 0
Examples
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, ... ...
Links
- Alois P. Heinz, Antidiagonals n = 0..34, flattened
Crossrefs
Programs
-
Maple
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); -
Mathematica
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 *)