A006026 Number of column-convex polyominoes with perimeter n. (Formerly M2924)

1, 3, 12, 54, 260, 1310, 6821, 36413, 198227, 1096259, 6141764, 34784432, 198828308, 1145544680, 6645621536, 38786564126, 227585926704, 1341757498470, 7944249448686, 47217102715624, 281615520373954, 1684957401786580, 10110628493454482, 60830401073611514

Offset

1,2

Comments

With offset 2, a(n) = number of directed column-convex polyominoes with directed-site perimeter = n. Directed means every cell (unit square) is reachable from the lower left cell, which is assumed to touch the origin. The directed-site perimeter is the number of unit squares in the first quadrant outside the polyomino but sharing at least one side with it. For example, the polyomino consisting of only one cell (with vertices (0,0),(1,0),(1,1),(0,1)) has directed-site perimeter = 2 due to the squares just above and to the right of it. - David Callan, Nov 29 2007

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Colin Barker, Table of n, a(n) for n = 1..1000

M.-P. Delest, Utilisation des Langages Algébriques et du Calcul Formel Pour le Codage et l'Enumeration des Polyominos, Ph.D. Dissertation, Université Bordeaux I, May 1987. [Scanned copy, with permission. A very large file.]

M.-P. Delest, Utilisation des Langages Algébriques et du Calcul Formel Pour le Codage et l'Enumeration des Polyominos, Ph.D. Dissertation, Université Bordeaux I, May 1987. (Annotated scanned copy of a small part of the thesis)

M.-P. Delest, Generating functions for column-convex polyominoes, J. Combin. Theory Ser. A 48 (1988), no. 1, 12-31.

Maylis P. Delest and Serge Dulucq, Enumeration of Directed Column-Convex Animals with a Given Perimeter and Area, Croatica Chemica Acta, 66 (1993), 59-80.

G. S. Joyce and A. J. Guttmann, Exact results for the generating function of directed column-convex animals on the square lattice, J. Physics A: Math. General 27 (1994) 4359-4367.

Formula

The g.f. A(x) = x + x^2 + 3x^3 + ... satisfies A^3 - 3A^2 + (1+2x)A - x = 0. - David Callan, Nov 29 2007

Mathematica

a[1]=1; a[2]=1; a[3]=3; a[n_]/; n>=4 := a[n] = ( 2(n-1)(21n-34)a[n-1] - (3n-8)(23n-43)a[n-2] + 16(n-3)(2n-7)a[n-3] )/(5(n-1)n); Table[a[n], {n, 10}] (* David Callan, Nov 29 2007 *)

Crossrefs

Cf. A006027, A259332, A259333.

Sequence in context: A362597 A125188 A054666 * A158826 A107264 A370441

Adjacent sequences: A006023 A006024 A006025 * A006027 A006028 A006029

Keyword

nonn,easy

Author

Simon Plouffe

Extensions

Delest thesis provided by M.-P. Delest and scanned by Simon Plouffe, Jan 16 2016

Status

approved