A000228 -id:A000228 - cp's OEIS frontend

A385266 Number of edge-connected components of polygonal cells in the basketweave tiling up to rotation and reflection of the tiling.

1, 2, 2, 10, 34, 166, 777, 4053, 21225, 114594, 624242, 3442399, 19121661, 106964679, 601639326, 3400619170, 19301719485, 109962791254

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

Each square cell in the basketweave tiling is edge-connected to four rectangular cells.

Peter Kagey, Illustration of the a(4)=34 polyforms with 4 cells.
Peter Kagey, Illustration of the basketweave tiling.

A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square), A344211 (rhombitrihexagonal), A344213 (truncated trihexagonal), A383908 (snub trihexagonal), A385265 (pinwheel).

A057783 Building block is 2 hexagons side-by-side; sequence gives number of pieces (polydohexes) that can be formed from n such pairs of hexagons.

Original entry on oeis.org

1, 6, 74, 1257, 25379, 544108, 12037738

Offset: 1

N. J. A. Sloane, Oct 29 2000

Computed by Brendan Owen.

Andrew Clarke, Other Polyforms
Sean A. Irvine, Java program (github)

Cf. A000228, A000105, A000665.

Link updated by William Rex Marshall, Dec 16 2009

a(7) from Sean A. Irvine, Jul 02 2022

A126026 Conjectured upper bound on area of the convex hull of any edge-to-edge connected system of regular unit hexagons (n-polyhexes).

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 10, 13, 17, 20, 24, 28, 33, 38, 43, 49, 55, 61, 68, 75, 82, 90, 97, 106, 114, 123, 133, 142, 152, 162, 173, 184, 195, 207, 219, 231, 244, 257, 270, 284, 297, 312, 326, 341, 357, 372, 388, 404, 421, 438, 455, 473, 491, 509, 528, 547, 566

Offset: 0

Jonathan Vos Post, Feb 27 2007

Kurz proved the polyomino equivalent of this conjecture as A122133 and abstracts: "In this article we prove a conjecture of Bezdek, Brass and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of n unit hypercubes in the d-dimensional Euclidean space. For d=2 we enumerate the extremal polyominoes and determine the set of possible areas of the convex hull for each n."

			a(10) = 24 because floor((10^2 + 14*10/3 + 1)/6) = floor(24.6111111) = 24.

Colin Barker, Table of n, a(n) for n = 0..1000
Sascha Kurz, Convex hulls of polyominoes, arXiv:math/0702786 [math.CO], Feb 26 2007. See conjecture 2, p. 12.
Eric Weisstein's World of Mathematics, Polyhex.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000228, A036359, A002216, A005963, A001998, A018190, A001207, A057973, A122133.

Table[Floor[(n^2+14n/3+1)/6],{n,0,80}] (* Harvey P. Dale, Apr 11 2012 *)

concat(0, Vec(x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)) + O(x^50))) \\ Colin Barker, Oct 13 2016

a(n) = (n^2 + 14*n/3 + 1)\6 \\ Charles R Greathouse IV, Oct 13 2016

a(n) = floor((n^2 + 14*n/3 + 1)/6).

G.f.: x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)). - Colin Barker, Oct 13 2016

More terms from Harvey P. Dale, Apr 11 2012

Offset changed to 0 by Colin Barker, Oct 13 2016

A131488 a(n) is the number of polyhexes with n edges, including inner edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 1, 5, 0, 1, 3, 6, 12, 3, 4, 14, 26, 39, 10, 25, 70, 116, 139, 67, 152, 347, 514, 567, 414, 884, 1744, 2408, 2561, 2498, 4967

Offset: 1

Tanya Khovanova, Jul 28 2007

An n-celled polyhex with perimeter p has (6n+p)/2 edges. The maximum number of edges in an n-celled polyhex is 5n+1.

Given Clarke's table T(p,n), a(n) is an antidiagonal sum selecting entries in a (1,3)-leaper's moves. - R. J. Mathar, Feb 23 2021

			a(31) = T(p=26,A=6) + T(p=20,A=7) = 36+3 = 39. a(34) = T(p=26,A=7) + T(p=20,A=8) = 69+1 = 70. a(35) = 107+9. a(36) = 118+21. a(41) = 411+155+1. a(44) = 1621 +123. a(45) = 1986+420+2. a(46) = 1489+1046+26. - _R. J. Mathar_, Feb 23 2021

Andrew Clarke, Isoperimetrical Polyhexes

Cf. A000228: Number of hexagonal polyominoes (or planar polyhexes) with n cells. A057779: Hexagonal polyominoes (or polyhexes, A000228) with perimeter 2n. A038142: Number of planar cata-polyhexes with n cells. A131487: analog for square tiling.

Extended to a(48). - R. J. Mathar, Feb 23 2021

A178778 Partial sums of walks of length n+1 on a tetrahedron A001998.

Original entry on oeis.org

1, 3, 7, 17, 42, 112, 308, 882, 2563, 7565, 22449, 66979, 200204, 599514, 1796350, 5385764, 16150725, 48442327, 145307291, 435892341, 1307617966, 3922765316, 11768118792, 35304090646, 105911740487, 317734424289, 953201678533, 2859602644103, 8578803149328

Offset: 0

Jonathan Vos Post, Dec 26 2010

The subsequence of primes begins 3, 7, 17, no more through a(27).

			a(5) = 1 + 2 + 4 + 10 + 25 + 70 = 112.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-12,21,-9).

Cf. A001998, A036359, A002216, A005963, A000228, A001997, A001444, A038766.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (6*x^3-4*x^2-2*x+1)/((x-1)^2*(3*x-1)*(3*x^2-1)) )); // G. C. Greubel, Jan 24 2019

CoefficientList[Series[(6*x^3-4*x^2-2*x+1)/((x-1)^2*(3*x-1)*(3*x^2-1)), {x,0,30}], x] (* G. C. Greubel, Jan 24 2019 *)

Vec((1-2*x-4*x^2+6*x^3)/((1-x)^2*(1-3*x)*(1-3*x^2)) + O(x^50)) \\ Colin Barker, May 17 2016

((6*x^3-4*x^2-2*x+1)/((x-1)^2*(3*x-1)*(3*x^2-1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 24 2019

a(n) = Sum_{i=0..n} (if i mod 2 = 0 then ((3^((i-2)/2)+1)/2)^2 else 3^((i-3)/2)+(1/4)*(3^(i-2)+1)).
G.f.: (6*x^3-4*x^2-2*x+1) / ((x-1)^2*(3*x-1)*(3*x^2-1)). - Colin Barker, Apr 20 2013
From Colin Barker, May 17 2016: (Start)
a(n) = (-7+3^(1+n)+3^(1/2*(-1+n))*(9-9*(-1)^n+5*sqrt(3)+5*(-1)^n*sqrt(3))+2*(1+n))/8.
a(n) = (2*n + 10*3^(n/2) + 3^(n+1) - 5)/8 for n even.
a(n) = (2*n + 3^(n+1) + 2*3^((n+3)/2) - 5)/8 for n odd.
a(n) = 5*a(n-1) - 4*a(n-2) - 12*a(n-3) + 21*a(n-4) - 9*a(n-5) for n>4.
(End)

A325936 a(n) is the number of creatures that can be made from exactly n Palago tiles.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 2, 9, 13, 37, 81, 205, 512, 1335, 3404, 8922, 23592, 62630, 167622

Offset: 1

Peter Kagey, Sep 09 2019

A "creature" in Palago is a closed shape of a single color across several Palago tiles, perhaps with one or more holes.
A Palago tile is a hexagonal tile with four regions of alternating colors. See links for illustrations.
Creatures are counted up to rotation, reflection, and swapping colors.

Cameron Browne, Counting Creatures
Code Golf Stack Exchange, Counting creatures on a hexagonal tiling

Cf. A000228.

A333018 Number of free heptagonal polyforms with n cells on the heptagonal tiling of the hyperbolic plane.

Original entry on oeis.org

1, 1, 1, 3, 10, 44, 249, 1513, 9992, 68305, 480748, 3450793, 25186583

Offset: 0

Peter Kagey, Mar 05 2020

The heptagonal tiling is represented by Schläfli symbol {7,3}.
This sequence is to A259352 what A000228 is to A108070.
This sequence is computed from via program by Christian Sievers in the Code Golf Stack Exchange link.

Code Golf Stack Exchange, Impress Donald Knuth by counting polyominoes on the hyperbolic plane
Wikipedia, Heptagonal tiling
Wikipedia, Polyform

Analogs with different Schläfli symbols are A000105 ({4,4}), A000207 ({3,oo}), A000228 ({6,3}), A000577 ({3,6}), A005036 ({4,oo}), A119611 ({4,5}), A330659 ({3,7}), and A332930 ({4,6}).

Cf. A108070, A259352.

a(9)-a(12) from Ed Wynn, Feb 14 2021

A345076 Number of generalized polyforms on the elongated triangular tiling with n cells.

Original entry on oeis.org

1, 2, 3, 5, 13, 32, 96, 283, 907, 2929, 9787, 32939, 112476, 386230, 1336150, 4642930, 16208851, 56786242, 199614651, 703678568, 2487109359, 8811020024, 31281360326, 111272475650

Offset: 0

Drake Thomas, Jun 07 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 glide reflection) of the other.

			See the PDF in the links section.

Peter Kagey, The a(3) = 5 generalized polyforms on the elongated triangular tiling with 3 cells.
Wikipedia, Elongated Triangular Tiling

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), 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(15)-a(23) from Bert Dobbelaere, Jun 05 2025

A378014 Triangle read by rows: T(n,k) = number of free hexagonal polyominoes with n cells, where the maximum number of cells on any lattice line is k. The term "lattice line" here means a line running through the cell centers and midpoints of their sides.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 3, 15, 3, 1, 0, 5, 50, 23, 3, 1, 0, 1, 171, 126, 30, 4, 1, 0, 1, 506, 710, 187, 39, 4, 1, 0, 1, 1459, 3520, 1268, 270, 48, 5, 1, 0, 1, 3792, 16617, 7703, 1948, 364, 59, 5, 1, 0, 1, 9292, 72870, 45099, 12885, 2840, 488, 70, 6, 1

Offset: 1

Dave Budd, Nov 14 2024

The row sums are the total number of free hexagonal polyominoes with n cells.

			   |  k
 n |       1      2      3      4      5      6      7      8      9     10       Total
---------------------------------------------------------------------------------------
 1 |       1                                                                          1
 2 |       0      1                                                                   1
 3 |       0      2      1                                                            3
 4 |       0      4      2      1                                                     7
 5 |       0      3     15      3      1                                             22
 6 |       0      5     50     23      3      1                                      82
 7 |       0      1    171    126     30      4      1                              333
 8 |       0      1    506    710    187     39      4      1                      1448
 9 |       0      1   1459   3520   1268    270     48      5      1               6572
10 |       0      1   3792  16617   7703   1948    364     59      5      1       30490
The T(4,2)=4 hexagon polyominoes are:
  #         #        #   #      # #
   # #       # #      # #      # #
      #     #

Dave Budd, Python code for a hex lattice

Row sums are A000228.

Cf. A377941, A378015.

A057782 Building block is trapezoid formed from 3 equilateral triangles; sequence gives number of pieces (polytraps) that can be formed from n such trapezoids.

Original entry on oeis.org

1, 9, 94, 1552, 27285, 509805, 9783124

Offset: 1

N. J. A. Sloane, Oct 29 2000

Also known as "Polytriamonds" because the building block is the unique triamond (composite of three equilateral triangles joined edge-to-edge). - Aaron N. Siegel, May 23 2022

Computed by Brendan Owen.

Abaroth's World, Polyhes, Polyhalfhexes & Polytriamonds
Andrew Clarke, Other Polyforms

Cf. A000577, A000228, A000105, A056844.

Link updated by William Rex Marshall, Dec 16 2009

a(7) from Aaron N. Siegel, May 23 2022

cp's OEIS Frontend

A385266 Number of edge-connected components of polygonal cells in the basketweave tiling up to rotation and reflection of the tiling.

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A057783 Building block is 2 hexagons side-by-side; sequence gives number of pieces (polydohexes) that can be formed from n such pairs of hexagons.

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A126026 Conjectured upper bound on area of the convex hull of any edge-to-edge connected system of regular unit hexagons (n-polyhexes).

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Mathematica

PARI

PARI

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A131488 a(n) is the number of polyhexes with n edges, including inner edges.

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A178778 Partial sums of walks of length n+1 on a tetrahedron A001998.

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Magma

Mathematica

PARI

Sage

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A325936 a(n) is the number of creatures that can be made from exactly n Palago tiles.

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A333018 Number of free heptagonal polyforms with n cells on the heptagonal tiling of the hyperbolic plane.

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A345076 Number of generalized polyforms on the elongated triangular tiling with n cells.

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A378014 Triangle read by rows: T(n,k) = number of free hexagonal polyominoes with n cells, where the maximum number of cells on any lattice line is k. The term "lattice line" here means a line running through the cell centers and midpoints of their sides.

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