A121580 -id:A121580 - cp's OEIS frontend

A121582 Number of cells in column 2 of all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

0, 1, 7, 40, 252, 1837, 15259, 141798, 1455694, 16360387, 199845957, 2637020884, 37388864368, 566971338009, 9157693715407, 156975522127762, 2846305448882274, 54432896145210943, 1095019542858729769

Offset: 1

Emeric Deutsch, Aug 11 2006

nonn

a(n)=Sum(k*A121581(n,k),k=0..n-1).

			a(2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 1 cells in their second columns.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Harvey P. Dale, Table of n, a(n) for n = 1..400

Cf. A121580, A121581, A121584.

a:=proc(n) if n=1 then 0 elif n=2 then 1 else ((2*n-3)*a(n-1)-(n-1)*a(n-2)+(n-1)!*(n-2)*(n^2-3*n+4)/2)/(n-2) fi end: seq(a(n),n=1..22);

RecurrenceTable[{a[1]==0,a[2]==1,a[n]==((2n-3)a[n-1]-(n-1)a[n-2]+ (n-1)!(n-2) (n^2-3n+4)/2)/(n-2)},a,{n,20}] (* Harvey P. Dale, Oct 23 2012 *)

a(1)=0, a(2)=1, a(n)=[(2n-3)a(n-1)-(n-1)a(n-2)+(n-1)!(n-2)(n^2-3n+4)/2]/(n-2) for n>=3.
a(n) ~ n*n!/2. - Vaclav Kotesovec, Aug 15 2013
D-finite with recurrence (-49*n+454)*a(n) +(49*n^2-454*n-1328)*a(n-1) +(49*n^2+1553*n-1464)*a(n-2) +(-581*n^2+612*n+1035)*a(n-3) +(819*n^2-3682*n+4007)*a(n-4) -4*(84*n-169)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

A121584 Number of cells in columns 1 and 2 of all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

Original entry on oeis.org

1, 4, 18, 93, 569, 4074, 33336, 306035, 3111771, 34708944, 421407314, 5533007841, 78125977725, 1180594364966, 19012215609564, 325058642549919, 5880810783960431, 112243265407073100, 2254038189505807926

Offset: 1

Emeric Deutsch, Aug 11 2006

nonn

a(n)=Sum(A121583(n,k),k=1..2n-2) for n>=2. a(n)=A121580(n)+A121582(n)

			a(2)=4 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, each having a total of 2 cells in their first two columns.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Cf. A121580, A121582, A121583.

a[1]:=1: a[2]:=4: for n from 3 to 22 do a[n]:=((2*n-3)*a[n-1]-(n-1)*a[n-2])/(n-2)+(1/2)*(2*n^3-9*n^2+17*n-16)*(n-1)!/(n-2) od: seq(a[n],n=1..22);

a(1)=1, a(2)=4, a(n)=[(2n-3)a(n-1)-(n-1)a(n-2)]/(n-2) + (1/2)(2n^3-9n^2+17n-16)(n-1)!/(n-2) for n>=3.

Conjecture D-finite with recurrence 14*(-n+1)*a(n) +(14*n^2+1731*n-6995)*a(n-1) +3*(-577*n^2+480*n+7243)*a(n-2) +2*(2781*n^2-11952*n+6004)*a(n-3) +(-5987*n^2+36181*n-54220)*a(n-4) +2*(1071*n-3433)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

cp's OEIS Frontend

A121582 Number of cells in column 2 of all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula