A000105 -id:A000105 - cp's OEIS frontend

A317186 One of many square spiral sequences: a(n) = n^2 + n - floor((n-1)/2).

1, 2, 6, 11, 19, 28, 40, 53, 69, 86, 106, 127, 151, 176, 204, 233, 265, 298, 334, 371, 411, 452, 496, 541, 589, 638, 690, 743, 799, 856, 916, 977, 1041, 1106, 1174, 1243, 1315, 1388, 1464, 1541, 1621, 1702, 1786, 1871, 1959, 2048, 2140, 2233, 2329, 2426

Offset: 0

N. J. A. Sloane, Jul 27 2018

Draw a square spiral on a piece of graph paper, and label the cells starting at the center with the positive (resp. nonnegative) numbers. This produces two versions of the labeled square spiral, shown in the Example section below.
The spiral may proceed clockwise or counterclockwise, and the first arm of the spiral may be along any of the four axes, so there are eight versions of each spiral. However, this has no effect on the resulting sequences, and it is enough to consider just two versions of the square spiral (starting at 1 or starting at 0).
The present sequence is obtained by reading alternate entries on the X-axis (say) of the square spiral started at 1.
The cross-references section lists many sequences that can be read directly off the two spirals. Many other sequences can be obtained from them by using them to extract subsequences from other important sequences. For example, the subsequence of primes indexed by the present sequence gives A317187.
a(n) is also the number of free polyominoes with n + 4 cells whose difference between length and width is n. In this comment the length is the longer of the two dimensions and the width is the shorter of the two dimensions (see the examples of polyominoes). Hence this is also the diagonal 4 of A379625. - Omar E. Pol, Jan 24 2025
From John Mason, Feb 19 2025: (Start)
The sequence enumerates polyominoes of width 2 having precisely 2 horizontal bars. By classifying such polyominoes according to the following templates, it is possible to define a formula that reduces to the one below:
.
 OO  O  O
 O  OO OO
 O  O  O
 O  O  OO
 OO OO  O
.
(End)

			The square spiral when started with 1 begins:
.
  100--99--98--97--96--95--94--93--92--91
                                        |
   65--64--63--62--61--60--59--58--57  90
    |                               |   |
   66  37--36--35--34--33--32--31  56  89
    |   |                       |   |   |
   67  38  17--16--15--14--13  30  55  88
    |   |   |               |   |   |   |
   68  39  18   5---4---3  12  29  54  87
    |   |   |   |       |   |   |   |   |
   69  40  19   6   1---2  11  28  53  86
    |   |   |   |           |   |   |   |
   70  41  20   7---8---9--10  27  52  85
    |   |   |                   |   |   |
   71  42  21--22--23--24--25--26  51  84
    |   |                           |   |
   72  43--44--45--46--47--48--49--50  83
    |                                   |
   73--74--75--76--77--78--79--80--81--82
.
For the square spiral when started with 0, subtract 1 from each entry. In the following diagram this spiral has been reflected and rotated, but of course that makes no difference to the sequences:
.
   99  64--65--66--67--68--69--70--71--72
    |   |                               |
   98  63  36--37--38--39--40--41--42  73
    |   |   |                       |   |
   97  62  35  16--17--18--19--20  43  74
    |   |   |   |               |   |   |
   96  61  34  15   4---5---6  21  44  75
    |   |   |   |   |       |   |   |   |
   95  60  33  14   3   0   7  22  45  76
    |   |   |   |   |   |   |   |   |   |
   94  59  32  13   2---1   8  23  46  77
    |   |   |   |           |   |   |   |
   93  58  31  12--11--10---9  24  47  78
    |   |   |                   |   |   |
   92  57  30--29--28--27--26--25  48  79
    |   |                           |   |
   91  56--55--54--53--52--51--50--49  80
    |                                   |
   90--89--88--87--86--85--84--83--82--81
.
From _Omar E. Pol_, Jan 24 2025: (Start)
For n = 0 there is only one free polyomino with 0 + 4 = 4 cells whose difference between length and width is 0 as shown below, so a(0) = 1.
   _ _
  |_|_|
  |_|_|
.
For n = 1 there are two free polyominoes with 1 + 4 = 5 cells whose difference between length and width is 1 as shown below, so a(1) = 2.
   _ _     _ _
  |_|_|   |_|_|
  |_|_|   |_|_
  |_|     |_|_|
.
(End)

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for sequences related to polyominoes.

Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Filling in these two squares spirals with greedy algorithm: A274640, A274641.
Cf. also A317187.
Cf. A000105, A379625.

a[n_] := n^2 + n - Floor[(n - 1)/2]; Array[a, 50, 0] (* Robert G. Wilson v, Aug 01 2018 *)
LinearRecurrence[{2, 0, -2 , 1},{1, 2, 6, 11},50] (* or *)
CoefficientList[Series[(- x^3 - 2 * x^2 - 1) / ((x - 1)^3 * (x + 1)), {x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)

From Daniel Forgues, Aug 01 2018: (Start)
a(n) = (1/4) * (4 * n^2 + 2 * n + (-1)^n + 3), n >= 0.
a(0) = 1; a(n) = - a(n-1) + 2 * n^2 - n + 2, n >= 1.
a(0) = 1; a(1) = 2; a(2) = 6; a(3) = 11; a(n) = 2 * a(n-1) - 2 * a(n-3) + a(n-4), n >= 4.
G.f.: (- x^3 - 2 * x^2 - 1) / ((x - 1)^3 * (x + 1)). (End)
E.g.f.: ((2 + 3*x + 2*x^2)*cosh(x) + (1 + 3*x + 2*x^2)*sinh(x))/2. - Stefano Spezia, Apr 24 2024
a(n)+a(n+1)=A033816(n). - R. J. Mathar, Mar 21 2025
a(n)-a(n-1) = A042948(n), n>=1. - R. J. Mathar, Mar 21 2025

A144553 Number of chiral pairs of polyominoes with n cells that have precisely the symmetry group of order 4 generated by 90-degree rotations.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 12, 7, 0, 0, 44, 25, 0, 0, 165, 90, 0, 0, 603, 319, 0, 0, 2235, 1136, 0, 0, 8283, 4088, 0, 0, 30936, 14868, 0, 0, 116111, 54526, 0, 0, 438465, 201527, 0, 0, 1663720, 750169, 0, 0, 6342211, 2809931, 0, 0, 24273767

Offset: 1

N. J. A. Sloane, Jan 01 2009

nonn

The values for n>28 were produced by a set of programs, the most difficult of which is attached. There is no guarantee that the values are correct, although presumably Shirakawa has calculated them through a(45). The attached program can be altered to count only achiral polyominoes, and those results match those of A142886, which uses a very different method. The difficulties lie in determining each inner loop (A324408 and A324409) and in determining connections within the inner loop (bad_connection subroutine). The last bug I found in the program affected only polyominoes with 72 or more cells. - Robert A. Russell, May 23 2020
These are polyominoes of the regular tiling with Schläfli symbol {4,4}. In late August, 2021, John Mason informed me that there were errors for a(44) and higher. My error in a(44) was a copying error, but later entries were wrong because of my programming errors. After making corrections (see attached C++ program), our values now match. John uses a unique calculation of his own devising. Since it is quite different from Redelmeier's inner rings, the match gives us some confidence in the current values. - Robert A. Russell, Nov 01 2021
Polyominoes with precisely 90-degree symmetry centered about square centers and vertices are enumerated by A351142 and A234007 respectively. - John Mason, Feb 17 2022

			For a(8)=1, the polyomino is a central 2 X 2 square with one cell attached to each edge of that square. - _Robert A. Russell_, Nov 01 2021

Robert A. Russell, Table of n, a(n) for n = 1..95
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Robert A. Russell, C++ Program
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351142, A234007.
Cf. A324408, A324409 (inner rings).
Cf. A348848 (C90), A348849 (F90).

A006747 = Cases[Import["https://oeis.org/A006747/b006747.txt",
     "Table"], {, }][[All, 2]];
A006749 = Cases[Import["https://oeis.org/A006749/b006749.txt",
     "Table"], {, }][[All, 2]];
A030228 = Cases[Import["https://oeis.org/A030228/b030228.txt",
     "Table"], {, }][[All, 2]];
a[n_] := A030228[[n+1]] - A006747[[n]] - A006749[[n]];
Array[a, 43] (* Jean-François Alcover, Sep 09 2019, updated Aug 17 2022 *)

a(n) = A030228(n) - A006747(n) - A006749(n). - Jean-François Alcover, Sep 09 2019, after Andrew Howroyd in A030228.
a(n) = (A348848(n/4)+A348849(n)-A142886(n)) / 2, where the first two are F90 and C90 of the Shirakawa link. - Robert A. Russell, Nov 01 2021
a(n) = A351142(n) + A234007(n/4) if n is a multiple of 4, otherwise a(n) = A351142(n). - John Mason, Feb 17 2022

a(28) added by Andrew Howroyd, Dec 04 2018
a(29)-a(91) added by Robert A. Russell, May 23 2020
Warning: It seems that the C++ program and the Mathematica program produce different results. This means that the b-file, and possibly even the terms in the DATA lines, are suspect. - N. J. A. Sloane, Aug 17 2022
After John Mason's Apr 15 2023 correction to the b-file of A006749, the discrepancy disappeared. - Andrey Zabolotskiy, Jan 18 2024

A006747 Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 18, 19, 73, 73, 278, 283, 1076, 1090, 4125, 4183, 15939, 16105, 61628, 62170, 239388, 240907, 932230, 936447, 3641945, 3651618, 14262540, 14277519, 55987858, 55961118, 220223982, 219813564, 867835023, 865091976, 3425442681

Offset: 1

N. J. A. Sloane

nonn

This sequence gives the number of free polyominoes with symmetry group "R" in Redelmeier's notation. See his Tables 1 and 3, also the column "Rot" in Oliveira e Silva's table.

The rotation center of a polyomino with this symmetry may lie at the center of a square, the middle of an edge, or a vertex of a square. These subsets are enumerated by A351615, A234008 and A351616 respectively. - John Mason, Feb 17 2022, reformulated by Günter Rote, Oct 19 2023

			a(2) = 0 because the "domino" polyomino has symmetry group of order 4.
For n=3, the three-celled polyomino [ | | ] has group of order 4, and the polyomino
. [ ]
. [ | ]
has only reflective symmetry, so a(3) = 0.
a(4) = 1 because of (in Golomb's notation) the "skew tetromino".

S. W. Golomb, Polyominoes, Princeton Univ. Press, NJ, 1994.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Mason, Table of n, a(n) for n = 1..50
Tomás Oliveira e Silva, Enumeration of polyominoes
Tomás Oliveira e Silva, Numbers of polyominoes classified according to Redelmeier's symmetry classes (an extract from the previous link)
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.
Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351615, A234008, A351616.
Polyomino rings of length 2n with twofold rotational symmetry: A348402, A348403, A348404.

a(n) = A351615(n) + A234008(n/2) + A351616(n/2) for even n, otherwise a(n) = A351615(n). - John Mason, Feb 17 2022

Extended to n=28 by Tomás Oliveira e Silva
a(1)-a(3) prepended by Andrew Howroyd, Dec 04 2018
Edited by N. J. A. Sloane, Nov 28 2020
a(29)-a(36) from John Mason, Oct 16 2021

A006749 Number of asymmetric polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 1, 5, 20, 84, 316, 1196, 4461, 16750, 62878, 237394, 899265, 3422111, 13069026, 50091095, 192583152, 742560511, 2870523142, 11122817672, 43191285751, 168046076423, 654997492842, 2557223459805, 9999080270766, 39152997087077, 153511067364760

Offset: 1

N. J. A. Sloane

This sequence counts polyominoes whose symmetry group has order 1.

A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891-904.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Mason, Table of n, a(n) for n = 1..50 (This version corrects erroneous values from a(44) onwards in previous version.)
Tomás Oliveira e Silva, Enumeration of polyominoes
T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy). See page 21.
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.

a(n) + A259090(n) = A000105(n). - R. J. Mathar, Sep 29 2021

Extended to n=28 by Tomás Oliveira e Silva.

A056877 Number of polyominoes with n cells, symmetric about two orthogonal axes.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 4, 4, 8, 10, 15, 17, 30, 35, 60, 64, 117, 128, 236, 241, 459, 476, 937, 912, 1813, 1789, 3706, 3456, 7187, 6779, 14712, 13161, 28571, 25839, 58457, 50348, 113798, 98957, 232718, 193375, 453969, 380522, 927601, 745248, 1813219, 1468202, 3702063

Offset: 1

N. J. A. Sloane, Sep 03 2000

nonn

This sequence counts polyominoes with exactly the symmetry group D_4 generated by horizontal and vertical reflections.
The subset of (2n)-ominoes with edge centers in this set are enumerated by A346799(n). - Robert A. Russell, Dec 15 2021
Polyominoes centered about square centers and vertices are enumerated by A351190 and A351191 respectively. - John Mason, Feb 15 2022

			For a(8)=4, the four octominoes with exactly fourfold symmetry and axes of symmetry parallel to the edges of the cells are a row of eight cells, two adjacent rows of four cells, a row of four cells with another four cells adjacent to its center cells, and a row of four cells with another four cells adjacent to its end cells (but not in the original row):
  XXXXXXXX
.
   XXXX
   XXXX
.
   XX
  XXXX
   XX
.
  X  X
  XXXX
  X  X

Robert A. Russell, Table of n, a(n) for n = 1..81
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Wikipedia, Symmetries of polyominoes

Cf. A000105, A001168, A006746, A006748, A056878, A006747, A006749, A346799, A351190, A351191.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.

a(n) = A351190(n) + A346799(n/2) + A351191(n/4) if we accept the convention that Axxxxxx(y) = 0 for any noninteger y. - John Mason, Feb 15 2022

More terms from Robert A. Russell, Jan 16 2019

A006746 Number of axially symmetric polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 9, 23, 38, 90, 147, 341, 564, 1294, 2148, 4896, 8195, 18612, 31349, 70983, 120357, 271921, 463712, 1045559, 1792582, 4034832, 6950579, 15619507, 27023509, 60638559, 105320716, 236006955, 411364068, 920626423, 1609836928

Offset: 1

N. J. A. Sloane

nonn

Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid. - Joshua Zucker, Mar 08 2008

The line of reflective symmetry may pass through the center of a square or a vertex of a square. These subsets are enumerated by A349328 and A349329 respectively. - John Mason, Feb 17 2022

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

John Mason, Table of n, a(n) for n = 1..50
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A030227, A361625.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A349328, A349329.

a(n) = A349328(n) + A349329(n/2) for even n, otherwise a(n) = A349328(n). - John Mason, Feb 17 2022

Extended to n=28 by Tomás Oliveira e Silva

A056878 Number of polyominoes with n cells, symmetric about diagonal 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 3, 3, 5, 6, 14, 9, 20, 20, 56, 32, 80, 64, 224, 114, 315, 217, 863, 397, 1234, 751, 3331, 1400, 4816, 2632, 12815, 4973, 18792, 9349, 49400, 17810, 73338, 33557, 190643, 64309, 286368, 121511, 737532, 233891, 1119215, 443271, 2859154

Offset: 1

N. J. A. Sloane, Sep 03 2000

nonn

The sequence refers to those polyominoes having reflective symmetry on both diagonals, consequent 180-degree rotational symmetry, but without 90-degree rotational symmetry. Such polyominoes with rotational symmetry symmetry centered about square centers and vertices are enumerated by A351159 and A351160 respectively. - John Mason, Feb 17 2022

			For a(7)=1, the heptomino with exactly fourfold symmetry and axes of symmetry parallel to the diagonals of the cells is composed of two 2 X 2 squares with one cell in common. For a(8)=1, the octomino is composed of a 2 X 2 square and the four cells adjacent to two nonadjacent cells of that square.

Robert A. Russell, Table of n, a(n) for n = 1..87
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351159, A351160.

a(n) = A351159(n) + A351160(n/2) for even n, otherwise a(n) = A351159(n). - John Mason, Feb 17 2022

More terms from Robert A. Russell, Jan 18 2019

A142886 Number of polyominoes with n cells that have the symmetry group D_8.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 12, 7, 0, 0, 20, 11, 0, 0, 45, 20, 0, 0, 80, 36, 0, 0, 173, 65, 0, 0, 310, 117, 0, 0, 664, 216, 0, 0, 1210, 396, 0, 0, 2570, 736, 0, 0, 4728, 1369, 0, 0, 9976, 2558, 0, 0, 18468, 4787, 0, 0, 38840

Offset: 0

N. J. A. Sloane, Jan 01 2009

nonn

This is the largest possible symmetry group that a polyomino can have.

Polyominoes with such symmetry centered about square centers and vertices are enumerated by A351127 and A346800 respectively. - John Mason, Feb 16 2022

			The monomino has eight-fold symmetry. The tetromino with eight-fold symmetry is four cells in a square. The pentomino with eight-fold symmetry is a cell and its four adjacent cells.

Robert A. Russell, Table of n, a(n) for n = 0..163
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Index entries for sequences related to groups

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351127, A346800.

Cf. A376971 (polycubes with full symmetry).

a(n) = A351127(n) + A346800(n/4) if n is a multiple of 4, otherwise a(n) = A351127(n). - John Mason, Feb 16 2022

Name corrected by Wesley Prosser, Sep 06 2017
a(28) added by Andrew Howroyd, Dec 04 2018
More terms from Robert A. Russell, Jan 13 2019

A006748 Number of diagonally symmetric polyominoes with n cells.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 7, 5, 26, 22, 91, 79, 326, 301, 1186, 1117, 4352, 4212, 16119, 15849, 60174, 60089, 226146, 228426, 854803, 872404, 3247207, 3342579, 12389106, 12850662, 47448984, 49544820, 182338754, 191529007, 702807040, 742163178, 2716205709, 2882119756

Offset: 1

N. J. A. Sloane

nonn

This sequence counts polyominoes with exactly the symmetry group of order 2 generated by a single reflection about a diagonal.

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

John Mason, Table of n, a(n) for n = 1..50 (terms up to a(47) from Robert A. Russell)
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Toshihiro Shirakawa, Harmonic Magic Square, pp 3-4: Enumeration of Polyominoes considering the symmetry, April 2012.

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.

a(n) = (A346800(n) - A142886(n)) / 2 - A056878(n). - Robert A. Russell, Aug 25 2021

Extended to n=28 by Tomás Oliveira e Silva

A343909 Number of generalized polyforms on the tetrahedral-octahedral honeycomb with n cells.

Original entry on oeis.org

1, 2, 1, 4, 9, 44, 195, 1186, 7385, 49444, 337504, 2353664, 16608401, 118432965, 851396696, 6163949361, 44896941979

Offset: 0

Drake Thomas and Peter Kagey, May 03 2021

This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection, or a combination thereof) of the other.

			For n = 1, the a(1) = 2 polyforms are the tetrahedron and the octahedron.
For n = 2, the a(2) = 1 polyform is a tetrahedron and an octahedron connected at a face.
For n = 3, there are a(3) = 4 polyforms with 3 cells:
  - 3 consisting of one octahedron with two tetrahedra, and
  - 1 consisting of two octahedra and one tetrahedron.
For n = 4, there are a(4) = 9 polyforms with 4 cells:
  - 3 with one octahedron and three tetrahedra,
  - 5 with two octahedra and three octahedra, and
  - 1 with three octahedra and one tetrahedron.
For n = 5, there are a(5) = 44 polyforms with 5 cells:
  - 6 with one octahedron and four tetrahedra,
  - 24 with two octahedra and three tetrahedra,
  - 13 with three octahedra and two tetrahedra, and
  - 1 with four octahedra and one tetrahedron.

Peter Kagey, Animation of the a(4) = 9 polyforms with 4 cells.
Peter Kagey, Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb, Mathematics Stack Exchange.
Wikipedia, Tetrahedral-octahedral honeycomb

Row sums of A365970.

Analogous for other honeycombs/tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A038119 (cubical), A068870 (tesseractic), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(11)-a(16) from Bert Dobbelaere, Jun 10 2025

cp's OEIS Frontend

A317186 One of many square spiral sequences: a(n) = n^2 + n - floor((n-1)/2).

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A144553 Number of chiral pairs of polyominoes with n cells that have precisely the symmetry group of order 4 generated by 90-degree rotations.

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A006747 Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).

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A006749 Number of asymmetric polyominoes with n cells.

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A056877 Number of polyominoes with n cells, symmetric about two orthogonal axes.

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A006746 Number of axially symmetric polyominoes with n cells.

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A056878 Number of polyominoes with n cells, symmetric about diagonal 2.

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A142886 Number of polyominoes with n cells that have the symmetry group D_8.

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