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A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n k-ominoes of shape I; 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, 13, 1, 1, 1, 1, 1, 41, 1, 1, 1, 1, 1, 19, 281, 1, 1, 1, 1, 1, 1, 57, 1183, 1, 1, 1, 1, 1, 1, 26, 121, 6728, 1, 1, 1, 1, 1, 1, 1, 75, 783, 31529, 1, 1, 1, 1, 1, 1, 1, 34, 154, 2861, 167089, 1, 1, 1, 1, 1, 1, 1, 1, 95, 269, 8133, 817991, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

A(n,n) = A034856(n+2) for n>=2.

LINKS

Alois P. Heinz, Antidiagonals n = 0..35, flattened

Wikipedia, Polyomino

EXAMPLE

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,     13,    1,    1,   1,   1,   1,  1, ...

  1, 1,     41,   19,    1,   1,   1,   1,  1, ...

  1, 1,    281,   57,   26,   1,   1,   1,  1, ...

  1, 1,   1183,  121,   75,  34,   1,   1,  1, ...

  1, 1,   6728,  783,  154,  95,  43,   1,  1, ...

  1, 1,  31529, 2861,  269, 190, 117,  53,  1, ...

  1, 1, 167089, 8133, 1732, 325, 229, 141, 64, ...

MAPLE

b:= proc(n, l) option remember; local d, k; d:= nops(l)/3;

      if n=0 then 1

    elif min(l[])>0 then (m->b(n-m, map(x->x-m, l)))(min(l[]))

    else for k while l[k]>0 do od;

         `if`(n<d, 0, b(n, subsop(k=d, l)))+

         `if`(d=1 or k>2*d+1 or max(l[k..k+d-1][])>0, 0,

          b(n, [l[1..k-1][], 1$d, l[k+d..3*d][]]))

      fi

    end:

A:= (n, k)-> `if`(k=0, 1, b(n, [0$3*k])):

seq(seq(A(n, d-n), n=0..d), d=0..12);

MATHEMATICA

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/3, k}, Which[n == 0, 1,  Min[l] > 0, Function[{m}, b[n-m, l-m]][Min[l]], True, For[k=1, l[[k]] > 0 , k++]; If[n<d, 0, b[n, ReplacePart[l, k -> d]]] + If[d == 1 || k > 2d + 1 || Max[l[[k ;; k + d - 1]]] > 0,  0,  b[n, Join[l[[1 ;; k-1]], Array[1&, d],  l[[k+d ;; 3*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 3k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Jan 30 2015, after Alois P. Heinz *)

CROSSREFS

Columns k=0+1,2-10 give: A000012, A028468, A251073, A251074, A247218, A251075, A251076, A251077, A251078, A251079.

Cf. A034856, A250662.

Sequence in context: A180265 A165400 A181154 * A332018 A010227 A010228

Adjacent sequences:  A251069 A251070 A251071 * A251073 A251074 A251075

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Nov 29 2014

STATUS

approved

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Last modified August 5 08:27 EDT 2021. Contains 346464 sequences. (Running on oeis4.)