A374794
Number of deco polyominoes of height 2n and vertical height n.
Original entry on oeis.org
1, 1, 10, 216, 8181, 489753, 43073059, 5251140144, 847811871333, 175006259417547, 44939475107574752, 14046429669829943012, 5249989348656458769520, 2312011774544840687484876, 1184766852578716585055014620, 698927709348312453031204116720
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..232
- Elena Barcucci, Sara Brunetti and Francesco Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
- Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
A121693
Number of deco polyominoes of height n and vertical height 3 (i.e., having 3 rows).
Original entry on oeis.org
0, 0, 1, 12, 57, 216, 741, 2412, 7617, 23616, 72381, 220212, 666777, 2012616, 6062421, 18236412, 54807537, 164619216, 494250861, 1483539012, 4452189897, 13359715416, 40085437701, 120268896012, 360831853857, 1082545893216
Offset: 1
- E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
- E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
- Index entries for linear recurrences with constant coefficients, signature (6,-11,6).
A121694
Sum of the vertical heights (i.e., number of rows) of all deco polyominoes of height n.
Original entry on oeis.org
1, 3, 12, 61, 377, 2734, 22671, 211035, 2175754, 24592551, 302295925, 4014475756, 57277225309, 873819665135, 14195291340656, 244657733062761, 4459137940238245, 85694418205589534, 1731893273528613811, 36721566227335477047, 815098440677104096866
Offset: 1
a(2)=3 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 1 and 2 rows.
- E. Barcucci, S. Brunetti, and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
- E. Barcucci, A. Del Lungo, and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
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with(linalg): a:=proc(i,j) if i=j then i elif i>j then 1 else 0 fi end: p:=proc(Q) local n,A,b,w,QQ: n:=degree(Q): A:=matrix(n,n,a): b:=j->coeff(Q,t,j): w:=matrix(n,1,b): QQ:=multiply(A,w): sort(expand(add(QQ[k,1]*t^k,k=1..n)+t*Q)): end: P[1]:=t: for n from 2 to 22 do P[n]:=p(P[n-1]) od: seq(subs(t=1,diff(P[n],t)),n=1..22);
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(* T is A121692 *)
T[n_, k_] := T[n, k] = Which[k == 1, 1, k == n, 1, k > n, 0, True, k*T[n-1, k] + 2*T[n-1, k-1] + Sum[T[n-1, j], {j, 1, k-2}]];
a[n_] := Sum[k*T[n, k], {k, 1, n}];
Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Aug 20 2024 *)
Showing 1-3 of 3 results.
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