

A006026


Number of columnconvex polyominoes with perimeter n.
(Formerly M2924)


4



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

1,2


COMMENTS

With offset 2, a(n) = number of directed columnconvex polyominoes with directedsite perimeter = n. Directed means every cell (unit square) is reachable from the lower left cell, which is assumed to touch the origin. The directedsite 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 directedsite 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 columnconvex polyominoes, J. Combin. Theory Ser. A 48 (1988), no. 1, 1231.
G. S. Joyce and A. J. Guttmann, Exact results for the generating function of directed columnconvex animals on the square lattice, J. Physics A: Math. General 27 (1994) 43594367.


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(n1)(21n34)a[n1]  (3n8)(23n43)a[n2] + 16(n3)(2n7)a[n3] )/(5(n1)n); Table[a[n], {n, 10}] (* David Callan, Nov 29 2007 *)


CROSSREFS

Sequence in context: A055835 A125188 A054666 * A158826 A107264 A200740
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



