A059716 -id:A059716 - cp's OEIS frontend

A001207 Number of fixed hexagonal polyominoes with n cells.

1, 3, 11, 44, 186, 814, 3652, 16689, 77359, 362671, 1716033, 8182213, 39267086, 189492795, 918837374, 4474080844, 21866153748, 107217298977, 527266673134, 2599804551168, 12849503756579, 63646233127758, 315876691291677, 1570540515980274, 7821755377244303, 39014584984477092, 194880246951838595, 974725768600891269, 4881251640514912341, 24472502362094874818, 122826412768568196148, 617080993446201431307, 3103152024451536273288, 15618892303340118758816, 78679501136505611375745

Offset: 1

N. J. A. Sloane

A. J. Guttmann, ed., Polygons, Polyominoes and Polycubes, Springer, 2009, p. 477. (Table 16.9 has 46 terms of this sequence.)
W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..46 (from reference by A. J. Guttmann)
Moa Apagodu, Counting hexagonal lattice animals, arXiv:math/0202295 [math.CO], 2002-2009.
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016.
M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers, Discrete Mathematics, Volume 258, Issues 1-3, 6 December 2002, Pages 235-274.
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.
Stephan Mertens, Markus E. Lautenbacher, Counting lattice animals: a parallel attack, J. Statist. Phys. 66 (1992), no. 1-2, 669-678.
H. Redelmeier, Emails to N. J. A. Sloane, 1991
M. F. Sykes, M. Glen. Percolation processes in two dimensions. I. Low-density series expansions, J. Phys A 9 (1) (1976) 87.
Markus Voege and Anthony J. Guttmann, On the number of hexagonal polyominoes, Theoretical Computer Sciences, 307(2) (2003), 433-453. (Table 2 has 35 terms of this sequence.)

Cf. A000228 (free), A006535 (one-sided).

Cf. A121220 (simply connected), A059716 (column convex).

3 more terms and reference from Achim Flammenkamp, Feb 15 1999

More terms from Markus Voege (markus.voege(AT)inria.fr), Mar 25 2004

A167012 Number of Level 2 hexagonal polyominoes with cheesy blocks and n cells.

Original entry on oeis.org

1, 3, 11, 44, 186, 810, 3582, 15952, 71242, 318441, 1423411, 6360809, 28415254, 126900911, 566604462, 2529439891, 11290673434, 50394458326, 224918228462, 1003813933351, 4479953995624, 19993503244811, 89228022987483, 398209768217607

Offset: 1

Jonathan Vos Post, Oct 26 2009

From Table 1, p.24, of Feretic. By level 0 cheesy polyominoes, and so too by level 0 polyominoes with cheesy blocks, Feretic appears to mean the usual column-convex polyominoes (A059716). See the paper for his definition.

Ray Chandler, Table of n, a(n) for n = 1..100
Svjetlan Feretic, Polyominoes with nearly convex columns: A model with semidirected blocks, Math. Commun. 15 (2010), 77--97, arXiv:0910.4780v1 [math.CO].
Wikipedia, Polyhex
Index entries for linear recurrences with constant coefficients, signature (16, -107, 391, -850, 1108, -797, 169, 266, -317, -159, 913, -1081, 672, -446, 268, -7, 158, -404, 222, -42, 70, -34).

Cf. A059716, A167011, A167013.

LinearRecurrence[{16,-107,391,-850,1108,-797,169,266,-317,-159,913,-1081,672,-446,268,-7,158,-404,222,-42,70,-34},{1,3,11,44,186,810,3582,15952,71242,318441,1423411,6360809,28415254,126900911,566604462,2529439891,11290673434,50394458326,224918228462,1003813933351,4479953995624,19993503244811},24] (* Ray Chandler, Jul 16 2015 *)

G.f.: (x*(1 - 13*x + 70*x^2 - 202*x^3 + 336*x^4 - 317*x^5 + 143*x^6 + 18*x^7 - 84*x^8 + 11*x^9 + 227*x^10 - 375*x^11 + 267*x^12 - 165*x^13 + 134*x^14 - 21*x^15 + 4*x^16 - 124*x^17 + 98*x^18 - 12*x^19 + 28*x^20 - 16*x^21)) / (1 - 16*x + 107*x^2 - 391*x^3 + 850*x^4 - 1108*x^5 + 797*x^6 - 169*x^7 - 266*x^8 + 317*x^9 + 159*x^10 - 913*x^11 + 1081*x^12 - 672*x^13 + 446*x^14 - 268*x^15 + 7*x^16 - 158*x^17 + 404*x^18 - 222*x^19 + 42*x^20 - 70*x^21 + 34*x^22).

Edited by Ralf Stephan, Feb 07 2014

Extended by Ray Chandler, Jul 16 2015

A167013 Number of Level 3 hexagonal polyominoes with cheesy blocks and n cells.

Original entry on oeis.org

1, 3, 11, 44, 186, 812, 3614, 16259, 73558, 333683, 1515454, 6885303, 31283654, 142121322, 645545957, 2931714681, 13312277095, 60440946141, 274391188445, 1245601594285, 5654137180147, 25664803641528, 116492672036579, 528751598530367

Offset: 1

Jonathan Vos Post, Oct 26 2009

nonn

From Table 1, p.24, of Feretic. By level 0 cheesy polyominoes, and so too by level 0 polyominoes with cheesy blocks, Feretic appears to mean the usual column-convex polyominoes (A059716). See the paper for his definition.

Ray Chandler, Table of n, a(n) for n = 1..200
Svjetlan Feretic, Polyominoes with nearly convex columns: A model with semidirected blocks, Math. Commun. 15 (2010), 77--97, arXiv:0910.4780v1 [math.CO].
Wikipedia, Polyhex
Index entries for linear recurrences with constant coefficients, signature (27, -334, 2515, -12906, 47836, -132248, 276956, -438796, 508406, -365771, -36865, 648120, -1344653, 1932847, -2126787, 1632701, -408884, -1117382, 2223607, -2392085, 1636807, -418146, -665251, 1211688, -1191386, 838060, -416174, 41907, 323733, -664097, 810808, -657803, 319442, -14159, -120746, 95202, -22341, 7930, -47294, 74720, -62640, 19120, 28394, -46822, 21864, 18416, -20930, -6617, 14093, -982, -5867, 2682, 642, -608, 88, -12).

Cf. A059716, A167011, A167012.

G.f.: (x*(1 - 24*x + 264*x^2 - 1766*x^3 + 8033*x^4 - 26297*x^5 + 63860*x^6 - 116445*x^7 + 157849*x^8 - 148533*x^9 + 61825*x^10 + 99443*x^11 - 308464*x^12 + 519182*x^13 - 655900*x^14 + 618461*x^15 - 344081*x^16 - 101610*x^17 + 519331*x^18 - 707969*x^19 + 601249*x^20 - 284943*x^21 - 68043*x^22 + 297023*x^23 - 346370*x^24 + 265550*x^25 - 140577*x^26 + 31503*x^27 + 64681*x^28 - 166424*x^29 + 234520*x^30 - 218182*x^31 + 130432*x^32 - 29144*x^33 - 33391*x^34 + 38482*x^35 - 12237*x^36 - 2050*x^37 - 6144*x^38 + 18593*x^39 - 21514*x^40 + 11634*x^41 + 3351*x^42 - 13907*x^43 + 12096*x^44 + 2302*x^45 - 8825*x^46 + 570*x^47 + 4681*x^48 - 1695*x^49 - 1519*x^50 + 1290*x^51 + 64*x^52 - 224*x^53 + 44*x^54 - 12*x^55)) / (1 - 27*x + 334*x^2 - 2515*x^3 + 12906*x^4 - 47836*x^5 + 132248*x^6 - 276956*x^7 + 438796*x^8 - 508406*x^9 + 365771*x^10 + 36865*x^11 - 648120*x^12 + 1344653*x^13 - 1932847*x^14 + 2126787*x^15 - 1632701*x^16 + 408884*x^17 + 1117382*x^18 - 2223607*x^19 + 2392085*x^20 - 1636807*x^21 + 418146*x^22 + 665251*x^23 - 1211688*x^24 + 1191386*x^25 - 838060*x^26 + 416174*x^27 - 41907*x^28 - 323733*x^29 + 664097*x^30 - 810808*x^31 + 657803*x^32 - 319442*x^33 + 14159*x^34 + 120746*x^35 - 95202*x^36 + 22341*x^37 - 7930*x^38 + 47294*x^39 - 74720*x^40 + 62640*x^41 - 19120*x^42 - 28394*x^43 + 46822*x^44 - 21864*x^45 - 18416*x^46 + 20930*x^47 + 6617*x^48 - 14093*x^49 + 982*x^50 + 5867*x^51 - 2682*x^52 - 642*x^53 + 608*x^54 - 88*x^55 + 12*x^56).

Edited by Ralf Stephan, Feb 07 2014

Extended by Ray Chandler, Jul 16 2015

A167011 Number of Level 1 hexagonal polyominoes with cheesy blocks and n cells.

Original entry on oeis.org

1, 3, 11, 44, 184, 784, 3363, 14451, 62097, 266716, 1145074, 4914448, 21087401, 90472315, 388129627, 1665025084, 7142592112, 30639836360, 131436162099, 563822359859, 2418629133001, 10375190596724, 44506436288882, 190919170388912, 818985577308225, 3513200788519075

Offset: 1

Jonathan Vos Post, Oct 26 2009

From Table 1, p.24, of Feretic. By level 0 cheesy polyominoes, and so too by level 0 polyominoes with cheesy blocks, Feretic appears to mean the usual column-convex polyominoes (A059716). See the paper for his definition.

Svjetlan Feretic, Polyominoes with nearly convex columns: A model with semidirected blocks, Math. Commun. 15 (2010), 77--97, arXiv:0910.4780v1 [math.CO].
Wikipedia, Polyhex
Index entries for linear recurrences with constant coefficients, signature (9,-27,32,-13,3,1).

Cf. A059716, A167012, A167013.

LinearRecurrence[{9,-27,32,-13,3,1},{1,3,11,44,184,784},26] (* Ray Chandler, Jul 16 2015 *)
Rest[CoefficientList[Series[x*(-1+6x-11x^2+6x^3-2x^4)/(-1+9x-27x^2+32x^3-13x^4+3x^5+x^6),{x,0,26}],x]] (* Ray Chandler, Jul 16 2015 *)

G.f.: x(-1+6x-11x^2+6x^3-2x^4)/(-1+9x-27x^2+32x^3-13x^4+3x^5+x^6).

Edited by Ralf Stephan, Feb 07 2014

A187077 Number of row-convex polyplets with n cells.

Original entry on oeis.org

1, 4, 18, 83, 385, 1788, 8305, 38575, 179170, 832189, 3865253, 17952864, 83385309, 387298083, 1798875698, 8355202169, 38807241321, 180247221864, 837190686169, 3888482927823, 18060759310562, 83886449530197, 389625723579965

Offset: 1

David Bevan, Mar 03 2011

nonn

Equivalent to a sequence of row-convex polyhexes (A059716).

			a(3) = 18 = A006770(3)-2 omits the two 3-plets with non-convex rows (V and inverted V).

Cf. A006770 (all fixed polyplets); A059716 (row-convex polyhexes); A001169 (row-convex polyominoes).

a[n_]:={1,4,18,83}[[n]]/;n<5; a[n_]:=a[n]=7a[n-1]-13a[n-2]+10a[n-3]-2a[n-4]; Array[a, 23]

G.f.: -((x(x-1)^3)/(1-7x+13x^2-10x^3+2x^4)).

a(n) = 7a(n-1)-13a(n-2)+10a(n-3)-2a(n-4) for n > 4.

A068091 Number of board-pair-pile hexagonal polyominoes with n cells.

Original entry on oeis.org

1, 3, 11, 44, 186, 814, 3648, 16611, 76437, 354112, 1647344, 7682237, 35873310, 167625690, 783470179, 3662035980, 17115684065, 79986841677, 373759118882, 1746296080947

Offset: 1

Moa Apagodu, Mar 22 2002 and Oct 31 2002

			This sequence first diverges from A059716 at n = 4. a(4) is 2 greater than A059716(4) because a(4) counts the following 2 fixed polyhexes of 4 cells that contain a column with two contiguous blocks of cells:
     _    _
   _/ \  / \_
  / \_/  \_/ \
  \_/      \_/
  / \_    _/ \
  \_/ \  / \_/
    \_/  \_/
This sequence first diverges from A001207 at n = 7. a(7) is 4 less than A001207(7) because a(7) does not count the following 4 fixed polyhexes of 7 cells that contain a column with more than two contiguous blocks of cells:
     _    _        _        _
   _/ \  / \_    _/ \      / \_
  / \_/  \_/ \  / \_/      \_/ \
  \_/      \_/  \_/          \_/
  / \_    _/ \  / \_        _/ \
  \_/ \  / \_/  \_/ \_    _/ \_/
  / \_/  \_/ \    \_/ \  / \_/
  \_/      \_/      \_/  \_/
  / \_    _/ \     _/ \  / \_
  \_/ \  / \_/    / \_/  \_/ \
    \_/  \_/      \_/      \_/

Moa Apagodu (formerly Mohamud Mohammed) and Stirling Chow, Counting hexagonal lattice animals confined to a strip, arXiv:math/0202295v5 [math.CO], 2009. See Section 7.
Moa Apagodu, Counting Hexagonal Lattice Animals Using Umbral-Transfer-Matrix Method [dead link]
Moa Apagodu, Maple programs.

Cf. A001170, A001207, A059716, A157608.

The sequence is generated by a Maple program that accompanies the paper "Counting Hexagonal Lattice Animals using Umbral-Transfer-Matrix Method (UTMM)"

A157608 Array read by antidiagonals, giving number of fixed hexagonal polyominoes of height up to n/2 and with hexagonal cell count k.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 3, 6, 2, 0, 0, 1, 3, 10, 11, 2, 0, 0, 1, 3, 11, 25, 19, 2, 0, 0, 1, 3, 11, 37, 61, 32, 2, 0, 0, 1, 3, 11, 43, 111, 142, 53, 2, 0, 0, 1, 3, 11, 44, 153, 320, 323, 87, 2, 0, 0, 1, 3, 11, 44, 177, 514, 896, 723, 142, 2, 0, 0

Offset: 1

Jonathan Vos Post, Mar 02 2009

			The array begins:
================================================
n=1 | 0 | 0 |  0 |  0 |   0 |   0 |   0 |    0 |
n=2 | 1 | 0 |  0 |  0 |   0 |   0 |   0 |    0 |
n=3 | 1 | 2 |  2 |  2 |   2 |   2 |   2 |    2 |
n=4 | 1 | 3 |  6 | 11 |  19 |  32 |  53 |   87 |
n=5 | 1 | 3 | 10 | 25 |  61 | 142 | 323 |  723 |
n=6 | 1 | 3 | 11 | 37 | 111 | 320 | 896 | 2461 |
================================================

Moa Apagodu (formerly Mohamud Mohammed) and Stirling Chow, Counting hexagonal lattice animals confined to a strip, arXiv:math/0202295v5 [math.CO], 2009. See Table 1.
Moa Apagodu, Maple programs.

Cf. A001170, A001207, A059716, A068091, A308359.

T(4, 4) = 11:
                                               
   / \/ \ / \/ \  / \         / \   / \   / \
  / \/ \/ \/ \/ \ \/ \/ \ / \/ \/ / \/ \ / \/ \
  \/ \/     \/ \/   \/ \/ \/ \/  / \/ \/ \/ \/ \
                          \/     \/    \/             \/
                              _
   / \  / \/ \  / \/ \   / \  / \
  / \/ \ \/ \/  \/ \/ / \/  \/ \_
  \/ \/   \/ \  / \/  / \/ \  / \/ \
    \/       \/  \/    \/ \/  \/ \_/

T(n, k) = A001207(k) for n >= 2*k. - Andrey Zabolotskiy, Aug 31 2024

Definition not clear to me! "Height" refers to the lattice or to the polyominoes? - N. J. A. Sloane, Mar 14 2009

Name clarified and more terms added by Andrey Zabolotskiy, Aug 24 2024

A225114 Number of skew partitions of n whose diagrams have no empty rows and columns.

Original entry on oeis.org

1, 1, 3, 9, 28, 87, 272, 850, 2659, 8318, 26025, 81427, 254777, 797175, 2494307, 7804529, 24419909, 76408475, 239077739, 748060606, 2340639096, 7323726778, 22915525377, 71701378526, 224349545236, 701976998795, 2196446204672, 6872555567553, 21503836486190, 67284284442622, 210528708959146

Offset: 0

Joerg Arndt, Apr 29 2013

nonn

A skew partition S of size n is a pair of partitions [p1,p2] where p1 is a partition of the integer n1, p2 is a partition of the integer n2, p2 is an inner partition of p1, and n=n1-n2. We say that p1 and p2 are respectively the inner and outer partitions of S. A skew partition can be depicted by a diagram made of rows of cells, in the same way as a partition. Only the cells of the outer partition p1 which are not in the inner partition p2 appear in the picture. [from the Sage manual, see links]

			The a(4)=28 skew partitions of 4 are
01:  [[4], []]
02:  [[3, 1], []]
03:  [[4, 1], [1]]
04:  [[2, 2], []]
05:  [[3, 2], [1]]
06:  [[4, 2], [2]]
07:  [[2, 1, 1], []]
08:  [[3, 2, 1], [1, 1]]
09:  [[3, 1, 1], [1]]
10:  [[4, 2, 1], [2, 1]]
11:  [[3, 3], [2]]
12:  [[4, 3], [3]]
13:  [[2, 2, 1], [1]]
14:  [[3, 3, 1], [2, 1]]
15:  [[3, 2, 1], [2]]
16:  [[4, 3, 1], [3, 1]]
17:  [[2, 2, 2], [1, 1]]
18:  [[3, 3, 2], [2, 2]]
19:  [[3, 2, 2], [2, 1]]
20:  [[4, 3, 2], [3, 2]]
21:  [[1, 1, 1, 1], []]
22:  [[2, 2, 2, 1], [1, 1, 1]]
23:  [[2, 2, 1, 1], [1, 1]]
24:  [[3, 3, 2, 1], [2, 2, 1]]
25:  [[2, 1, 1, 1], [1]]
26:  [[3, 2, 2, 1], [2, 1, 1]]
27:  [[3, 2, 1, 1], [2, 1]]
28:  [[4, 3, 2, 1], [3, 2, 1]]

Sage Development Team, Skew Partitions, Sage Reference Manual.

\\ The following program is significantly faster.
A225114(n)=
{
    my( C=vector(n, j, 1) );
    my(m=n, z, t, ret);
    while ( 1,  /* for all compositions C[1..m] of n */
\\        print( vector(m, n, C[n] ) ); /* print composition */
        t = prod(j=2,m, min(C[j-1], C[j]) + 1 );  /* A225114 */
\\        t = prod(j=2,m, min(C[j-1], C[j]) + 0 );  /* A006958 */
\\        t = prod(j=2,m, C[j-1] + C[j] + 0 );  /* A059716 */
\\        t = prod(j=2,m, C[j-1] + C[j] + 1 );  /* A187077 */
\\        t = sum(j=2,m, C[j-1] > C[j] );  /* A045883 */
        ret += t;
        if ( m<=1, break() ); /* last composition? */
        /* create next composition: */
        C[m-1] += 1;
        z = C[m];
        C[m] = 1;
        m += z - 2;
    );
    return(ret);
}
for (n=0, 30, print1(A225114(n),", "));
\\ Joerg Arndt, Jul 09 2013

[SkewPartitions(n).cardinality() for n in range(16)]

Conjectured g.f.: 1/(2 - 1/(1 - x/(1 - x/(1 - x^2/(1 - x^2/(1 - x^3/(1 - x^3/(1 - ...)))))))). - Mikhail Kurkov, Sep 03 2024

Edited by Max Alekseyev, Dec 22 2015

A319322 Number of row convex polyhexes with n cells.

Original entry on oeis.org

1, 3, 11, 44, 184, 784, 3370, 14544, 62862, 271804, 1175133, 5079516, 21951384, 94847528, 409767878, 1770177486

Offset: 1

David Bevan, Sep 18 2018

A polyhex is considered to be row convex if cells take contiguous positions in each row. (Multiple cells in a row are not connected.)

			The only pentahexes that are not row convex are the shallow V shape and its 180-degree rotation. So a(5) = A001207(5) - 2 = 184.

Adam Gruber, Java program to count lattice animals.

Cf. A001207 (fixed polyhexes), A059716 (column convex polyhexes).

cp's OEIS Frontend

A001207 Number of fixed hexagonal polyominoes with n cells.

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A167012 Number of Level 2 hexagonal polyominoes with cheesy blocks and n cells.

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A167013 Number of Level 3 hexagonal polyominoes with cheesy blocks and n cells.

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A167011 Number of Level 1 hexagonal polyominoes with cheesy blocks and n cells.

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A187077 Number of row-convex polyplets with n cells.

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A068091 Number of board-pair-pile hexagonal polyominoes with n cells.

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Maple

A157608 Array read by antidiagonals, giving number of fixed hexagonal polyominoes of height up to n/2 and with hexagonal cell count k.

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Maple

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A225114 Number of skew partitions of n whose diagrams have no empty rows and columns.

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PARI

Sage