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 A322494 Number A(n,k) of tilings of a k X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation; square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 8, 5, 1, 1, 1, 1, 8, 18, 18, 8, 1, 1, 1, 1, 13, 44, 68, 44, 13, 1, 1, 1, 1, 21, 107, 233, 233, 107, 21, 1, 1, 1, 1, 34, 257, 838, 1262, 838, 257, 34, 1, 1, 1, 1, 55, 621, 2989, 6523, 6523, 2989, 621, 55, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS The shapes of the tiles are: ._. ._. | | ._. | | | | ._. | |_. | |_._. | |_._._. |_| |___| |_____| |_______| ... . . The sequence of column k (or row k) satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0. LINKS Alois P. Heinz, Antidiagonals n = 0..23, flattened Wikipedia, Polyomino EXAMPLE A(3,3) = 8: ._____. ._____. ._____. ._____. ._____. ._____. ._____. ._____. |_|_|_| | |_|_| |_|_|_| |_| |_| |_|_|_| |_| |_| | |_|_| | | |_| |_|_|_| |___|_| | |_|_| |_|___| |_| |_| | |___| | |_|_| | |___| |_|_|_| |_|_|_| |___|_| |_|_|_| |_|___| |___|_| |_____| |_____|. . Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 2, 3, 5, 8, 13, 21, 34, ... 1, 1, 3, 8, 18, 44, 107, 257, 621, ... 1, 1, 5, 18, 68, 233, 838, 2989, 10687, ... 1, 1, 8, 44, 233, 1262, 6523, 34468, 181615, ... 1, 1, 13, 107, 838, 6523, 51420, 396500, 3086898, ... 1, 1, 21, 257, 2989, 34468, 396500, 4577274, 52338705, ... 1, 1, 34, 621, 10687, 181615, 3086898, 52338705, 888837716, ... MAPLE b:= proc(n, l) option remember; local k, m, r; if n=0 or l=[] then 1 elif min(l)>0 then (t-> b(n-t, map(h->h-t, l)))(min(l)) elif l[-1]=n then b(n, subsop(-1=[][], l)) else for k while l[k]>0 do od; r:= 0; for m from 0 while k+m<=nops(l) and l[k+m]=0 and n>m do r:= r+b(n, [l[1..k-1][], 1\$m, m+1, l[k+m+1..nops(l)][]]) od; r fi end: A:= (n, k)-> b(max(n, k), [0\$min(n, k)]): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA b[n_, l_] := b[n, l] = Module[{k, m, r}, Which[n == 0 || l == {}, 1, Min[l] > 0, Function[t, b[n-t, l-t]][Min[l]], l[[-1]] == n, b[n, ReplacePart[ l, -1 -> Nothing]], True, For[k=1, l[[k]] > 0, k++]; r = 0; For[m=0, k+m <= Length[l] && l[[k+m]] == 0 && n>m, m++, r = r + b[n, Join[l[[1 ;; k-1]], Array[1&, m], {m+1}, l[[k+m+1 ;; Length[l]]]]]]; r]]; A[n_, k_] := b[Max[n, k], Array[0&, Min[n, k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2018, after Alois P. Heinz *) CROSSREFS Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A322496, A322497, A322498, A322499, A322500, A322501, A322502, A322503. Main diagonal gives A322495. Cf. A226444. Sequence in context: A219924 A226444 A196929 * A258445 A129179 A120621 Adjacent sequences: A322491 A322492 A322493 * A322495 A322496 A322497 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 12 2018 STATUS approved

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Last modified March 1 18:30 EST 2024. Contains 370443 sequences. (Running on oeis4.)