A006027 -id:A006027 - cp's OEIS frontend

A005435 Number of column-convex polyominoes with perimeter 2n+2.

1, 2, 7, 28, 122, 558, 2641, 12822, 63501, 319554, 1629321, 8399092, 43701735, 229211236, 1210561517, 6432491192, 34364148528, 184463064936, 994430028087, 5381653402890, 29226425965907, 159227245772460, 870004781620093, 4766330416567254, 26176585256712224

Offset: 1

Simon Plouffe

			a(3)=7 because we have: the 2 X 2 square, the 3 X 1 and 1 X 3 rectangles and the four polyominoes obtained by removing any of the four cells of the 2 X 2 square.

S. Feretic and D. Svrtan, On the number of column-convex polyominoes with given perimeter and number of columns, Proc. 5th Conf. Formal Power Series and Algebraic Combinatorics, Florence, 1993, pp. 201-214.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..400
R. Brak, A. J. Guttmann and I. G. Enting, Exact solution of the row-convex perimeter generating function, J. Phys. A 23 (1990), 2319-2326.
M.-P. Delest, Generating functions for column-convex polyominoes, J. Combin. Theory Ser. A 48 (1988), no. 1, 12-31.
E. Duchi and S. Rinaldi, An object grammar for column-convex polyominoes, Annals of Combinatorics, 8 (2004), 27-36.
S. Feretic, A new way of counting the column-convex polyominoes by perimeter, Discrete Math., 180, 1998, 173-184.
Svjetlan Feretić, The perimeter generating function for nondirected diagonally convex polyominoes, arXiv:1907.09409 [math.CO], 2019. See Eq. (28).

Cf. A006027, A269228.

assume(y,positive): G:=((y^2 - 1)*( - 21 + 47*y^2 - 35*y^4 + 5*y^6) - 3*(y^2 - 1)^2*(1 + y^2)*sqrt(1 - 6*y^2 + y^4) - 9*sqrt(2)*(y^2 - 1)^2*sqrt((y^2 - 1)^2*(1 + y^2) - (y^2 - 1)*(1 + y^2)*sqrt(1 - 6*y^2 + y^4)) - sqrt(2)*y*(y^2 - 1)*(1 + y^2)*sqrt((y^2 - 1)^2*(1 + y^2) + (y^2 - 1)*(1 + y^2)*sqrt(1 - 6*y^2 + y^4)))/(72 - 152*y^2 + 92*y^4 - 8*y^6): Gser:=series(G,y=0,70): seq(coeff(Gser,y^(2*n+2)),n=1..31); # Emeric Deutsch, May 13 2006

$Assumptions = (y > 0); terms = 25; ((y^2 - 1)*(-21 + 47*y^2 - 35*y^4 + 5*y^6) - 3*(y^2 - 1)^2*(1 + y^2)*Sqrt[1 - 6*y^2 + y^4] - 9*Sqrt[2]*(y^2 - 1)^2*Sqrt[(y^2 - 1)^2*(1 + y^2) - (y^2 - 1)*(1 + y^2)*Sqrt[1 - 6*y^2 + y^4]] - Sqrt[2]*y*(y^2 - 1)*(1 + y^2)*Sqrt[(y^2 - 1)^2*(1 + y^2) + (y^2 - 1)*(1 + y^2)*Sqrt[1 - 6*y^2 + y^4]])/(72 - 152*y^2 + 92*y^4 - 8*y^6) + O[y]^(2 terms + 3) // Normal // Simplify // CoefficientList[#, y^2]& // #[[3 ;; terms + 2]]& (* Jean-François Alcover, May 15 2017, translated from Maple *)

See the g.f. in the Maple program (taken from the Brak et al. paper). It has been given previously, in a different form, in the Delest paper (p. 29). - Emeric Deutsch, May 13 2006

Corrected by Simon Plouffe.

More terms from Emeric Deutsch, May 13 2006

A006026 Number of column-convex polyominoes with perimeter n.

Original entry on oeis.org

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

Simon Plouffe

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

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

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.

Cf. A006027, A259332, A259333.

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 *)

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

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

A105450 a(n) = binomial(n+5,6) + binomial(n+3,3) + binomial(n+2,3) + binomial(n-1,1).

Original entry on oeis.org

0, 6, 22, 60, 142, 305, 607, 1134, 2008, 3396, 5520, 8668, 13206, 19591, 28385, 40270, 56064, 76738, 103434, 137484, 180430, 234045, 300355, 381662, 480568, 600000, 743236, 913932, 1116150, 1354387, 1633605, 1959262, 2337344, 2774398, 3277566, 3854620

Offset: 0

D. G. Rogers, May 07 2005

Number of directed column-convex polyominoes with perimeter 2(n+4) having n cells in the foundational column.

A051743 and this sequence form successive diagonals in an array that has as row sums the sequence A006027.

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A051743, A006027.

Table[Binomial[n+5,6]+Binomial[n+3,3]+Binomial[n+2,3]+ Binomial[n-1,1],{n,0,50}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,6,22,60,142,305,607},51] (* Harvey P. Dale, Jun 28 2011 *)

a(n)=n*(n^5+15*n^4+85*n^3+465*n^2+1354*n+2400)/720 \\ Charles R Greathouse IV, Oct 16 2015

a(0)=0, a(1)=6, a(2)=22, a(3)=60, a(4)=142, a(5)=305, a(6)= 607, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)- 7*a(n-6)+a(n-7). - Harvey P. Dale, Jun 28 2011

G.f.: (2*x^6-11*x^5+26*x^4-32*x^3+20*x^2-6*x)/(x-1)^7. - Harvey P. Dale, Jun 28 2011

A051743 a(n) = (1/24)n(n + 5)*(n^2 + n + 6).

Original entry on oeis.org

2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, 1377, 1833, 2394, 3075, 3892, 4862, 6003, 7334, 8875, 10647, 12672, 14973, 17574, 20500, 23777, 27432, 31493, 35989, 40950, 46407, 52392, 58938, 66079, 73850, 82287, 91427, 101308, 111969, 123450

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999

This is exactly the number of directed column-convex polyominoes. [Something is clearly missing from this sentence; as it stands, it makes no reference to the index n. - Jon E. Schoenfield, Dec 20 2016]

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-3)=coeff(charpoly(A,x),x^(n-4)). [Milan Janjic, Jan 24 2010]

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A005435, A006027, A105450.

Table[(n (n + 5) (n^2 + n + 6))/24, {n, 50}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {2, 7, 18, 39, 75}, 50]

Vec((x^3-3*x^2+3*x-2)/(x-1)^5 + O(x^50)) \\ G. C. Greubel, Dec 21 2016

a(n) = binomial(n+3, n-1) + binomial(n, n-1) = binomial(n+3, 4) + binomial(n, 1), n > 0.
From Harvey P. Dale, Nov 29 2011: (Start)
a(1)=2, a(2)=7, a(3)=18, a(4)=39, a(5)=75, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (x^3-3*x^2+3*x-2)/(x-1)^5. (End)
E.g.f.: (1/24)*(48*x + 36*x^2 + 12*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 21 2016

A259333 Triangle read by rows: T(n,k) = number of column-convex polyominoes with bond-perimeter 2*n+2 and k columns (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 37, 16, 1, 1, 25, 105, 106, 25, 1, 1, 36, 240, 446, 245, 36, 1

Offset: 1

N. J. A. Sloane, Jun 24 2015

			Triangle begins:
1,
1,1,
1,4,1,
1,9,9,1,
1,16,37,16,1,
1,25,105,106,25,1,
1,36,240,446,245,36,1,
...

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.] See Figure 9.
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.
S. Dulucq, Etude combinatoire de problèmes d'énumeration, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)

Row sums are A006027.

There is an explicit formula for T(n,k) - see Delest (1987), Theorem 24.

cp's OEIS Frontend

A005435 Number of column-convex polyominoes with perimeter 2n+2.

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A006026 Number of column-convex polyominoes with perimeter n.

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A105450 a(n) = binomial(n+5,6) + binomial(n+3,3) + binomial(n+2,3) + binomial(n-1,1).

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A051743 a(n) = (1/24)*n*(n + 5)*(n^2 + n + 6).

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A259333 Triangle read by rows: T(n,k) = number of column-convex polyominoes with bond-perimeter 2*n+2 and k columns (1 <= k <= n).

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Formula

A051743 a(n) = (1/24)n(n + 5)*(n^2 + n + 6).