A005435 -id:A005435 - cp's OEIS frontend

A006027 Number of directed column-convex polyominoes with perimeter 2n+2.

1, 1, 2, 6, 20, 71, 263, 1005, 3933, 15684, 63505, 260390, 1079019, 4511700, 19011521, 80653480, 344193353, 1476589475, 6364258163, 27545933212, 119676949397, 521739175908, 2281673067934, 10006784399183, 44002280467770, 193957104163645, 856853526774173

Offset: 1

Simon Plouffe

nonn

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..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.
M.-P. Delest and S. Dulucq, Enumeration of directed column-convex animals with given perimeter and area, Croat. Chem. Acta. 66 (1993), 59-80.
E. Duchi and S. Rinaldi, An object grammar for column-convex polyominoes, Annals of Combinatorics, 8 (2004), 27-36.
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)

Cf. A005435.

m = 30; A[_] = 0;
Do[A[x_] = (2(1-x) A[x]^2 - A[x]^3 + x^2 - x^3)/((1-x)(1-2x))+O[x]^m, {m}];
CoefficientList[1 + A[x]/x, x] (* Jean-François Alcover, Oct 05 2019 *)

G.f. A(x) = a(1)x^2 + a(2)x^3 + a(3)x^4 + ... satisfies the functional equation A^3 + 2(x-1)A^2 + (2x-1)(x-1)A + (x^2)(x-1) = 0. - D. G. Rogers, May 22 2005

More terms from D. G. Rogers and Emanuele Munarini, May 15 2005

A259364 a(n) = 18n^4(2n^3 - 23n^2 + 38*n - 18)^2.

Original entry on oeis.org

0, 18, 93312, 4737042, 51775488, 263351250, 807055488, 1609827282, 1934917632, 774840978, 691920000, 20514061458, 126428055552, 496767242322, 1543426109568, 4122612551250, 9879830396928, 21788831695122, 44962051370112, 87830997546258, 163819480320000

Offset: 0

Vincenzo Librandi, Jun 25 2015

M. P. Delest, Generating functions for column-convex polyominoes, J. Combin. Theory Ser. A 48 (1988), no. 1, pp. 12-31.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A001105, A005435.

[18*n^4*(2*n^3-23*n^2+38*n-18)^2: n in [0..20]];

A259364:=n->18*n^4*(2*n^3 - 23*n^2 + 38*n - 18)^2: seq(A259364(n), n=0..30); # Wesley Ivan Hurt, Apr 12 2017

Table[18 n^4 (2 n^3 - 23 n^2 + 38 n - 18)^2, {n, 0, 23}]

[2*(6*n^5-69*n^4+114*n^3-54*n^2)^2 for n in (0..20)] # Bruno Berselli, Jun 25 2015

G.f.: 18*x*(1 + 5173*x + 206200*x^2 + 266512*x^3 - 3390686*x^4 + 389794*x^5 + 10761232*x^6 + 5689720*x^7 + 580693*x^8 + 6561*x^9)/(1-x)^11.

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11).

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

A259395 a(n) = -3n^2(n-1)^4(n+1)(11n^3+49n^2-439*n+171).

Original entry on oeis.org

0, 0, 15228, 705024, 1885680, -66355200, -792382500, -4986842112, -22516232256, -81696522240, -252908835300, -693126720000, -1723987588752, -3961019252736, -8517765880260, -17315965900800, -33541737120000, -62298041352192, -111515651966916, -193198552634880

Offset: 0

Vincenzo Librandi, Jun 26 2015

M. P. Delest, Generating functions for column-convex polyominoes, J. Combin. Theory Ser. A 48 (1988), no. 1, pp. 12-31. See expression D in Theorem 16 page 29.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A001105, A005435, A259364.

[-3*n^2*(n-1)^4*(n+1)*(11*n^3+49*n^2-439*n+171): n in [0..20]];

A259395:=n->-3*n^2*(n-1)^4*(n+1)*(11*n^3+49*n^2-439*n+171): seq(A259395(n), n=0..25); # Wesley Ivan Hurt, Jun 29 2015

Table[-3 n^2 (n - 1)^4 (n + 1) (11 n^3 + 49 n^2 - 439 n + 171), {n, 0, 23}]
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,0,15228,705024,1885680,-66355200,-792382500,-4986842112,-22516232256,-81696522240,-252908835300},20] (* Harvey P. Dale, Jul 07 2025 *)

G.f.: 324*x^2*(47+1659*x - 15531*x^2 - 156895*x^3 - 216255*x^4 - 17547*x^5 + 31451*x^6 + 3471*x^7) / (1-x)^11.

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11).

A269228 Number of nondirected diagonally convex polyominoes with perimeter 2n + 2.

Original entry on oeis.org

1, 2, 7, 28, 122, 556, 2618, 12634, 62128, 310212, 1568495, 8014742, 41323641, 214719610, 1123244757, 5910863420, 31268459118, 166185855552, 886961294034, 4751819567488, 25545030878475, 137756210983218, 745003421378887, 4039670554117446, 21957581725458521

Offset: 1

Svjetlan Feretic, Jul 11 2016

nonn

The generating function satisfies an algebraic equation of degree eight. I computed that generating function using the "turbo Temperley" method.
The formula for the generating function is given in the enclosed Maple worksheet.
The most practical version of the "turbo Temperley" method was given in Bousquet-Mélou's paper cited below.
The first five terms are the same as in the sequence A005435.
A005435(n) is the number of column-convex polyominoes with perimeter 2n + 2.
A049124(n) is the number of directed diagonally convex polyominoes with perimeter 2n.

			a(7) = 2618, so there are 2618 nondirected diagonally convex polyominoes with perimeter 2*7 + 2 = 16.

Svjetlan Feretic, Table of n, a(n) for n = 1..100
M. Bousquet-Mélou, A method for the enumeration of various classes of column-convex polygons, Discrete Math. 154 (1996), 1-25.
Svjetlan Feretić, Maple worksheet with g.f.
Svjetlan Feretić, the first one hundred terms of the sequence A269228
Svjetlan Feretić, The perimeter generating function for nondirected diagonally convex polyominoes, arXiv:1907.09409 [math.CO], 2019.

Cf. A005435, A049124.

A259435 a(n) = 2(n-1)^6(n+1)^2(n^2+10n+1).

Original entry on oeis.org

2, 0, 450, 81920, 2077650, 22413312, 148531250, 716636160, 2763575010, 9017753600, 25850353122, 66816000000, 158678718770, 351151718400, 731985584850, 1449526034432, 2745436781250, 5000952545280, 8800799033090, 15019798118400, 24938174692242, 40392704000000

Offset: 0

Vincenzo Librandi, Jun 27 2015

This appears as the function alpha(n) in Delest, related to bar/bat theory; see section 3.

M. P. Delest, Generating functions for column-convex polyominoes, J. Combin. Theory Ser. A 48 (1988), no. 1, pp. 12-31. See expression E in Theorem 16 page 29.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A001105, A005435, A259364, A259395.

[2*(n-1)^6*(n+1)^2*(n^2+10*n+1): n in [0..30]];

A259435:=n->2*(n-1)^6*(n+1)^2*(n^2+10*n+1): seq(A259435(n), n=0..30); # Wesley Ivan Hurt, Jun 29 2015

Table[2 (n - 1)^6 (n + 1)^2 (n^2 + 10 n + 1), {n, 0, 30}]
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{2,0,450,81920,2077650,22413312,148531250,716636160,2763575010,9017753600,25850353122},30] (* Harvey P. Dale, Dec 29 2024 *)

a(n)=2*(n-1)^6*(n+1)^2*(n^2+10*n+1) \\ Charles R Greathouse IV, Jun 29 2015

[2*(n-1)^6*(n+1)^2*(n^2+10*n+1) for n in (0..30)] # Bruno Berselli, Jun 30 2015

G.f.: 2*(1 -11*x + 280*x^2 + 38320*x^3 + 600970*x^4 + 1994794*x^5 + 1444096*x^6 - 231320*x^7 - 207395*x^8 - 10935*x^9)/(1-x)^11.

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11).

A060379 Number of self-avoiding polygons on the 2-dimensional square lattice with perimeter 2n with at most 4 horizontal edges in each vertical cross-section.

Original entry on oeis.org

1, 2, 7, 28, 124, 588, 2938, 15266, 81770, 448698, 2510813, 14277838, 82286365, 479610362, 2822332127, 16745262798

Offset: 2

Doron Zeilberger, Apr 03 2001

			a(3) = 2 because there are 2 self-avoiding polygons of perimeter 2*3 with at most 4 horizontal edges per vertical cross-section.

Doron Zeilberger, The Umbral Transfer-Matrix Method. IV. Counting Self-AvoidingPolygons and walks; Local copy [Pdf file only, no active links]

Cf. A005435, A002931.

See Appendix 2 of the reference (a 7-page system of linear functional equations for 5 unknown generating functions, one of which is the desired generating function).

A259563 a(n) = 81n^3(n-1)^5(n+1)^2(n^2-6n+1)(n^3-79n^2+163n-81).

Original entry on oeis.org

0, 0, 2571912, 2472394752, 138662798400, 1666179072000, -4637478825000, -272992368918528, -3187483870330368, -23209723979366400, -126970182577359000, -566493158246400000, -2161675076294530368, -7278963158259007488, -22112928086617859400, -61611251010011136000

Offset: 0

Vincenzo Librandi, Jun 30 2015

M. P. Delest, Generating functions for column-convex polyominoes, J. Combin. Theory Ser. A 48 (1988), no. 1, pp. 12-31. See expression F in Theorem 16 page 29.
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A001105, A005435, A259364, A259395, A259435.

[81*n^3*(n-1)^5*(n+1)^2*(n^2-6*n+1)*(n^3-79*n^2+163*n-81): n in [0..20]];

Table[81 n^3 (n-1)^5 (n+1)^2 (n^2 - 6 n + 1) (n^3 - 79 n^2 + 163 n - 81), {n, 0, 23}]
LinearRecurrence[{16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1},{0,0,2571912,2472394752,138662798400,1666179072000,-4637478825000,-272992368918528,-3187483870330368,-23209723979366400,-126970182577359000,-566493158246400000,-2161675076294530368,-7278963158259007488,-22112928086617859400,-61611251010011136000},20] (* Harvey P. Dale, Sep 02 2024 *)

G.f.: 52488*x^2*(49 + 46320*x + 1894016*x^2 - 4899760*x^3 - 305530185*x^4 - 1372006208*x^5 - 1287460720*x^6 + 1418574528*x^7 + 2422309735*x^8 + 1000206160*x^9 + 139190784*x^10 + 5654000*x^11 + 37281*x^12)/(1-x)^16.

a(n) = 16*a(n-1) - 120*a(n-2) + 560*a(n-3) - 1820*a(n-4) + 4368*a(n-5) - 8008*a(n-6) + 11440*a(n-7) - 12870*a(n-8) + 11440*a(n-9) - 8008*a(n-10) + 4368*a(n-11) - 1820*a(n-12) + 560*a(n-13) - 120*a(n-14) + 16*a(n-15) - a(n-16).

cp's OEIS Frontend

A006027 Number of directed column-convex polyominoes with perimeter 2n+2.

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A259364 a(n) = 18*n^4*(2*n^3 - 23*n^2 + 38*n - 18)^2.

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

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A259395 a(n) = -3*n^2*(n-1)^4*(n+1)*(11*n^3+49*n^2-439*n+171).

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A269228 Number of nondirected diagonally convex polyominoes with perimeter 2n + 2.

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A259435 a(n) = 2*(n-1)^6*(n+1)^2*(n^2+10*n+1).

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Magma

Maple

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A060379 Number of self-avoiding polygons on the 2-dimensional square lattice with perimeter 2n with at most 4 horizontal edges in each vertical cross-section.

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A259563 a(n) = 81*n^3*(n-1)^5*(n+1)^2*(n^2-6*n+1)*(n^3-79*n^2+163*n-81).

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A259364 a(n) = 18n^4(2n^3 - 23n^2 + 38*n - 18)^2.

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

A259395 a(n) = -3n^2(n-1)^4(n+1)(11n^3+49n^2-439*n+171).

A259435 a(n) = 2(n-1)^6(n+1)^2(n^2+10n+1).

A259563 a(n) = 81n^3(n-1)^5(n+1)^2(n^2-6n+1)(n^3-79n^2+163n-81).