A247218 -id:A247218 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Nov 29 2014

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

			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, ...

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

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

Cf. A034856, A250662.

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`(n2*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);

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]]] + 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-François Alcover, Jan 30 2015, after Alois P. Heinz *)

A247117 Number of tilings of a 10 X n rectangle using 2n pentominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 17, 28, 41, 56, 144, 317, 609, 1060, 1716, 3324, 6713, 13188, 24624, 43620, 80464, 153645, 296025, 562097, 1037921, 1920661, 3600832, 6820873, 12920804, 24211457, 45173688, 84493668, 158848825, 299451277, 562923960, 1055117520, 1976475968

Offset: 0

Alois P. Heinz, Nov 19 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A003520 (5 X n), A247218 (15 X n).

Column k=5 of A250662.

gf:= -(x^10+x^8-x^6-2*x^5-x^4-x^3+1) *(x-1)^4 *(x^4+x^3+x^2+x+1)^4 / (x^35 +x^33 -2*x^31 -7*x^30 -2*x^29 -6*x^28 +x^27 +9*x^26 +22*x^25 +8*x^24 +15*x^23 -4*x^22 -15*x^21 -39*x^20 -12*x^19 -20*x^18 +6*x^17 +10*x^16 +45*x^15 +8*x^14 +19*x^13 -4*x^12 -4*x^11 -33*x^10 -6*x^9 -10*x^8 +x^7 -3*x^6 +12*x^5 +x^3 +x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: see Maple program.

cp's OEIS Frontend

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.

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