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 A233308 Number A(n,k) of tilings of a k X k X n box using k*n bricks of shape k X 1 X 1; square array A(n,k), n>=0, k>=1, read by antidiagonals. 7
 1, 1, 1, 1, 2, 1, 1, 2, 9, 1, 1, 2, 4, 32, 1, 1, 2, 4, 21, 121, 1, 1, 2, 4, 8, 92, 450, 1, 1, 2, 4, 8, 45, 320, 1681, 1, 1, 2, 4, 8, 16, 248, 1213, 6272, 1, 1, 2, 4, 8, 16, 93, 1032, 4822, 23409, 1, 1, 2, 4, 8, 16, 32, 668, 3524, 18556, 87362, 1, 1, 2, 4, 8, 16, 32, 189, 3440, 13173, 70929, 326041, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Alois P. Heinz, Antidiagonals n = 0..17, flattened FORMULA A(n,k) = 2^n = A000079(n) for k>n. A(n,n) = A068156(n) for n>1. EXAMPLE Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, ... 1, 2, 2, 2, 2, 2, ... 1, 9, 4, 4, 4, 4, ... 1, 32, 21, 8, 8, 8, ... 1, 121, 92, 45, 16, 16, ... 1, 450, 320, 248, 93, 32, ... 1, 1681, 1213, 1032, 668, 189, ... 1, 6272, 4822, 3524, 3440, 1832, ... 1, 23409, 18556, 13173, 13728, 11976, ... MAPLE b:= proc(n, l) option remember; local d, t, k; d:= isqrt(nops(l)); if max(l[])>n then 0 elif n=0 then 1 elif min(l[])>0 then t:=min(l[]); b(n-t, map(x->x-t, l)) else for k while l[k]>0 do od; b(n, subsop(k=d, l))+ `if`(irem(k, d)=1 and {seq(l[k+j], j=1..d-1)}={0}, b(n, [seq(`if`(h-k=0, 1, l[h]), h=1..nops(l))]), 0)+ `if`(k<=d and {seq(l[k+d*j], j=1..d-1)}={0}, b(n, [seq(`if`(irem(h-k, d)=0, 1, l[h]), h=1..nops(l))]), 0) fi end: A:= (n, k)-> `if`(k>n, 2^n, b(n, [0\$k^2])): seq(seq(A(n, 1+d-n), n=0..d), d=0..11); MATHEMATICA b[n_, l_] := b[n, l] = Module[{d, t, k}, d= Sqrt[Length[l]]; Which[ Max[l]>n, 0, n==0, 1, Min[l]>0, t=Min[l]; b[n-t, l-t], True, k=Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k->d]]+ If[Mod[k, d]==1 && Union[ Table[ l[[k+j]], {j, 1, d-1}]] == {0}, b[n, Table[ If [h-k=0, 1, l[[h]] ], {h, 1, Length[l]}]], 0]+ If[k <= d && Union[ Table[ l[[k+d*j]], {j, 1, d-1}]] == {0}, b[n, Table[ If[ Mod[h-k, d] == 0, 1, l[[h]] ], {h, 1, Length[l]}]], 0] ] ]; a[n_, k_]:= If[k>n, 2^n, b[n, Array[0&, k^2]]]; Table[Table[a[n, 1+d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *) CROSSREFS Columns k=1-6 give: A000012, A006253, A233289, A233291, A233294, A233424. Diagonals include: A000079, A068156. Sequence in context: A082386 A336521 A356093 * A028306 A111259 A304195 Adjacent sequences: A233305 A233306 A233307 * A233309 A233310 A233311 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 07 2013 STATUS approved

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Last modified March 30 07:09 EDT 2023. Contains 361606 sequences. (Running on oeis4.)