A000228 -id:A000228 - cp's OEIS frontend

A001998 Bending a piece of wire of length n+1; walks of length n+1 on a tetrahedron; also non-branched catafusenes with n+2 condensed hexagons.

1, 2, 4, 10, 25, 70, 196, 574, 1681, 5002, 14884, 44530, 133225, 399310, 1196836, 3589414, 10764961, 32291602, 96864964, 290585050, 871725625, 2615147350, 7845353476, 23535971854, 70607649841, 211822683802, 635467254244, 1906400965570, 5719200505225, 17157599124190

Offset: 0

N. J. A. Sloane

The wire stays in the plane, there are n bends, each is R,L or O; turning the wire over does not count as a new figure.
Equivalently, walks of n+1 steps on a tetrahedron, visiting n+2 vertices, with n "corners"; the symmetry group is S4, reversing a walk does not count as different. Simply interpret R,L,O as instructions to turn R, turn L, or retrace the last step. Walks are not self-avoiding.
Also, it appears that a(n) gives the number of equivalence classes of n-tuples of 0, 1 and 2, where two n-tuples are equivalent if one can be obtained from the other by a sequence of operations R and C, where R denotes reversal and C denotes taking the 2's complement (C(x)=2-x). This has been verified up to a(19)=290585050. Example: for n=3 there are ten equivalence classes {000, 222}, {001, 100, 122, 221}, {002, 022, 200, 220}, {010, 212}, {011, 110, 112, 211}, {012, 210}, {020, 202}, {021, 102, 120, 201}, {101, 121}, {111}, so a(3)=10. - John W. Layman, Oct 13 2009
There exists a bijection between chains of n+2 hexagons and the above described equivalence classes of n-tuples of 0,1, and 2. Namely, for a given chain of n+2 hexagons we take the sequence of the numbers of vertices of degree 2 (0, 1, or 2) between the consecutive contact vertices on one side of the chain; switching to the other side we obtain the 2's complement of this sequence; reversing the order of the hexagons, we obtain the reverse sequence. The inverse mapping is straightforward. For example, to a linear chain of 7 hexagons there corresponds the 5-tuple 11111. - Emeric Deutsch, Apr 22 2013
If we treat two wire bends (or walks, or tuples) related by turning over (or reversing) as different in any of the above-given interpretations of this sequence, we get A007051 (or A124302). Also, a(n-1) is the sum of first 3 terms in n-th row of A284949, see crossrefs therein. - Andrey Zabolotskiy, Sep 29 2017
a(n-1) is the number of color patterns (set partitions) in an unoriented row of length n using 3 or fewer colors (subsets). - Robert A. Russell, Oct 28 2018
From Allan Bickle, Jun 02 2022: (Start)
a(n) is the number of (unlabeled) 3-paths with n+6 vertices. (A 3-path with order n at least 5 can be constructed from a 4-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to an existing 3-clique containing an existing 3-leaf.)
Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)
a(n) is also the number of distinct planar embeddings of the (n+1)-alkane graph (up to at least n=9, and likely for all n). - Eric W. Weisstein, May 21 2024

			There are 2 ways to bend a piece of wire of length 2 (bend it or not).
For n=4 and a(n-1)=10, the 6 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, and ABBC.  The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - _Robert A. Russell_, Oct 28 2018

A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 75.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70.
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [I think this reference does not mention this sequence. - N. J. A. Sloane, Aug 10 2006]
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).

Indranil Ghosh, Table of n, a(n) for n = 0..2092 (terms 0..500 from T. D. Noe)
A. T. Balaban, J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures, Tetrahedron 44(1) (1988), 221-228. See Eq. (6), p. 223.
A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, Tetrahedron 24 (1968), 2505-2516.
Christian Barrientos and Sarah Minion, On the Graceful Cartesian Product of Alpha-Trees, Theory and Applications of Graphs, Vol. 4: Iss. 1, Article 3, 2017. (It mentions this sequence on p. 7.)
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147.
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147 [annotated scanned copy].
Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of Molecular structure 376 (Issues 1-3) (1996), 495-505. See Table 2 on p. 501.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51.
R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51 [annotated scanned copy].
F. Harary and R. W. Robinson, Tapeworms, Unpublished manuscript, circa 1973. (Annotated scanned copy)
Thomas M. Liggett and Wenpin Tang, One-dependent hard-core processes and colorings of the star graph, arXiv:1804.06877 [math.PR], 2018.
L. Markenzon, O. Vernet, and P. R. da Costa Pereira, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Gyula Tasi and Fujio Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).
Eric Weisstein's World of Mathematics, Alkane Graph.
Eric Weisstein's World of Mathematics, Planar Embedding.
Index entries for sequences obtained by enumerating foldings
Index entries for linear recurrences with constant coefficients, signature (4,0,-12,9).

Cf. A036359, A002216, A005963, A000228, A001997, A001444, A038766, A007051.
Column 3 of A320750, offset by one. Column k = 0 of A323942, offset by two.
Cf. A124302 (oriented), A107767 (chiral), A182522 (achiral), with varying offsets.
Column 3 of A320750.
The numbers of unlabeled k-paths for k = 2..7 are given in A005418, A001998, A056323, A056324, A056325, and A345207, respectively.
The sequences above converge to A103293(n+1).

a:=[];; for n in [2..45] do if n mod 2 =0 then Add(a,((3^((n-2)/2)+1)/2)^2); else Add(a,  3^((n-3)/2)+(1/4)*(3^(n-2)+1)); fi; od; a; # Muniru A Asiru, Oct 28 2018

A001998 := proc(n) if n = 0 then 1 elif n mod 2 = 1 then (1/4)*(3^n+4*3^((n-1)/2)+1) else (1/4)*(3^n+2*3^(n/2)+1); fi; end;
A001998:=(-1+3*z+2*z**2-8*z**3+3*z**4)/(z-1)/(3*z-1)/(3*z**2-1); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence with an extra leading 1

a[n_?OddQ] := (1/4)*(3^n + 4*3^((n - 1)/2) + 1); a[n_?EvenQ] := (1/4)*(3^n + 2*3^(n/2) + 1); Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jan 25 2013, from formula *)
LinearRecurrence[{4,0,-12,9},{1,2,4,10},30] (* Harvey P. Dale, Apr 10 2013 *)
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=3; Table[Sum[StirlingS2[n,j]+Ach[n,j],{j,k}]/2,{n,40}] (* Robert A. Russell, Oct 28 2018 *)

Vec((1-2*x-4*x^2+6*x^3)/((1-x)*(1-3*x)*(1-3*x^2)) + O(x^50)) \\ Colin Barker, May 15 2016

a(n) = if n mod 2 = 0 then ((3^((n-2)/2)+1)/2)^2 else 3^((n-3)/2)+(1/4)*(3^(n-2)+1).
G.f.: (1-2*x-4*x^2+6*x^3) / ((1-x)*(1-3*x)*(1-3*x^2)). - Corrected by Colin Barker, May 15 2016
a(n) = 4*a(n-1)-12*a(n-3)+9*a(n-4), with a(0)=1, a(1)=2, a(2)=4, a(3)=10. - Harvey P. Dale, Apr 10 2013
a(n) = (1+3^n+3^(1/2*(-1+n))*(2-2*(-1)^n+sqrt(3)+(-1)^n*sqrt(3)))/4. - Colin Barker, May 15 2016
E.g.f.: (2*sqrt(3)*sinh(sqrt(3)*x) + 3*exp(2*x)*cosh(x) + 3*cosh(sqrt(3)*x))/6. - Ilya Gutkovskiy, May 15 2016
From Robert A. Russell, Oct 28 2018: (Start)
a(n-1) = (A124302(n) + A182522(n)) / 2 = A124302(n) - A107767(n-1) = A107767(n-1) + A182522(n).
a(n-1) = Sum_{j=1..k} (S2(n,j) + Ach(n,j)) / 2, where k=3 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n-1) = A057427(n) + A056326(n) + A056327(n). (End)
a(2*n) = A007051(n)^2; a(2*n+1) = A007051(n)*A007051(n+1). - Todd Simpson, Mar 25 2024

Offset and Maple code corrected by Colin Mallows, Nov 12 1999

Term added by Robert A. Russell, Oct 30 2018

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

A197465 Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

Original entry on oeis.org

1, 2, 2, 6, 8, 22, 42, 112, 252, 650, 1584, 4091, 10369, 26938, 69651, 182116, 476272, 1253067, 3302187, 8733551, 23142116, 61477564, 163612714, 436278921, 1165218495, 3117021788, 8349892686, 22397754046, 60153261611

Offset: 1

Joseph Myers, Oct 15 2011

See the link below for a definition of the tetrakis square tiling. When a square grid cell is divided into triangles, it must be divided dexter (\) or sinister (/) according to the parity of the grid cell.

			For n=3 there are 4 triaboloes.  Of these, 2 conform to the tetrakis grid.  Each of these 2 has a unique dissection into 6 tetrakis cells. - _George Sicherman_, Mar 25 2021

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Example illustrating a(4) = 6.
Wikipedia, Tetrakis square tiling

Cf. A197466, A197467, A006074.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

Name clarified by George Sicherman, Mar 25 2021
a(21)-a(26) from Aaron N. Siegel, May 18 2022
a(27)-a(29) from Bert Dobbelaere, Jun 04 2025

A343577 Number of generalized polyforms on the truncated square tiling with n cells.

Original entry on oeis.org

1, 2, 2, 7, 22, 93, 413, 2073, 10741, 57540, 312805, 1722483, 9564565, 53489304, 300840332, 1700347858, 9650975401

Offset: 0

Peter Kagey, Apr 20 2021

a(n) >= A343417(n), the number of (n-k)-polyominoes with k distinguished vertices.

Kadon Enterprises, Interesting solutions: Ochominoes.
Peter Kagey, Haskell program for computing sequence.
Peter Kagey, The a(3) = 7 generalized polyforms on the truncated square tiling with 3 cells.
Peter Kagey, The a(5) = 93 generalized polyforms on the truncated square tiling with 5 cells.
John Mason, Illustration of equivalence between truncated square polyforms and a mixture of crosses and squares.
Wikipedia, Truncated Square Tiling

Cf. A121197 (one-sided).

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).

a(11) from Drake Thomas, May 02 2021

a(12)-a(16) from John Mason, Mar 20 2022

A197159 Number of free poly-[3^4.6]-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 3, 8, 25, 80, 291, 1036, 3875, 14561, 55624, 213951, 830712, 3244355, 12747718, 50320659, 199491045, 793754027, 3168756843, 12687184463, 50932363171

Offset: 1

Joseph Myers, Oct 10 2011

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Example illustrating that a(3) = 8.
Wikipedia, Floret pentagonal tiling

Cf. A197160.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(16)-a(20) from Bert Dobbelaere, Jun 02 2025

A197459 Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

Original entry on oeis.org

1, 1, 3, 4, 12, 27, 78, 208, 635, 1859, 5726, 17526, 54620, 170479, 536714, 1694567, 5376764, 17110286, 54631302, 174879997, 561229678, 1805022806, 5817191196, 18781911278, 60744460580

Offset: 1

Joseph Myers, Oct 15 2011

[3.6.3.6] refers to the face configuration of the rhombille tiling. - Peter Kagey, Mar 01 2020
If we draw the short diagonals of each tile in the rhombille tiling, we get a subset of edges of the regular hexagonal grid; two edges are adjacent if and only if the corresponding rhombi are adjacent. These are polyedges where angles are constrained to 120 degrees. So there is a 1-to-1 correspondence with the subset of polyedges counted in A159867 after removing polyedges with angles of 60 and/or 180 degrees. - Joseph Myers, Jul 12 2020
These are also known as polytwigs, by association with their representation as polyedges. - Aaron N. Siegel, May 15 2022

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Abaroth's World, Illustration of a(1) through a(6)
Peter Kagey, Illustration for A197459(4) = 4.
R. J. Mathar, OEIS A197459
Wikipedia, Rhombille tiling

Cf. A019988, A159867, A197460, A197461.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(18)-a(22) from Aaron N. Siegel, May 15 2022

a(23)-a(25) from Bert Dobbelaere, Jun 04 2025

A197462 Number of free poly-[4.6.12]-tiles (holes allowed) with n cells (division into triangles is significant).

Original entry on oeis.org

1, 3, 3, 9, 14, 38, 74, 185, 414, 1026, 2440, 6077, 14926, 37454, 93749, 237035, 599815, 1526020, 3889117, 9944523, 25475398, 65416733, 168277945, 433705325, 1119610147, 2894928713

Offset: 1

Joseph Myers, Oct 15 2011

[4.6.12] refers to the face configuration of the kisrhombille tiling. - Peter Kagey, May 10 2021

Branko Gruenbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Illustration of a(4) = 9.
Wikipedia, Kisrhombille tiling

Cf. A197463, A197464.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(20)-a(26) from Aaron N. Siegel, Jun 03 2022

A197156 Number of free poly-[3^3.4^2]-tiles (polyhouses) (holes allowed) with n cells.

Original entry on oeis.org

1, 3, 5, 20, 56, 225, 819, 3333, 13336, 55231, 229146, 963284, 4068503, 17301000, 73893082, 317013121, 1364917667, 5896350458, 25545737979, 110968732581

Offset: 1

Joseph Myers, Oct 10 2011

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Example illustrating that a(3) = 5.
Brendan Owen, Polyhouses (gives a(1)-a(10)).
Wikipedia, Prismatic pentagonal tiling

Cf. A197157, A197158.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(16)-a(20) from Bert Dobbelaere, Jun 02 2025

A309159 Number of generalized polyforms on the snub square tiling with n cells.

Original entry on oeis.org

1, 2, 2, 4, 10, 28, 79, 235, 720, 2254, 7146, 22927, 74137, 241461, 790838, 2603210, 8604861, 28549166, 95027832, 317229779, 1061764660, 3562113987, 11976146355

Offset: 0

Peter Kagey and Peter Taylor, Jul 15 2019

nonn

The generalized polyforms counted by this sequence are "free", which means that they are counted up to rotation and reflection.

Peter Kagey, Counting generalized polyominoes
Wikipedia, Snub square tiling

Cf. A000105, A000228, A000577.

a(19)-a(22) from Christian Sievers

A258206 Number of strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotations and turning over.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 312, 744, 2291, 6186, 18714, 53793, 162565, 482416, 1467094, 4436536, 13594266, 41640513, 128564463, 397590126, 1236177615, 3852339237, 12053032356, 37802482958, 118936687722, 375079338476

Offset: 1

Antti Karttunen, May 31 2015

Differs from A057779 for the first time at n=12 as here a(12) = 97, one less than A057779(12) because this sequence excludes polyhexes with holes, the smallest which contains six hexagons in a ring, enclosing a hole of one hex, having thus perimeter of 18+6 = 24 (= 2*12) edges.
Differs from A258019 for the first time at n=13 as here a(13) = 312, one less than A258019(13) because this sequence counts only strictly non-overlapping and non-touching polyhex-patterns, while A258019(13) already includes one specimen of helicene-like self-reaching structures.
If one counts these structures by the number of hexagons (instead of perimeter length), one obtains sequence 1, 1, 3, 7, 22, 81, ... (A018190).
a(n) is also the number of 2n-step 2-dimensional closed self-avoiding paths on honeycomb lattice, reduced for symmetry. - Luca Petrone, Jan 08 2016

S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Theory of Coronoid Hydrocarbons, Springer-Verlag, 1991. See sections 4.7 Annulene and 6.5 Annulenes.

Hugo Pfoertner, Illustration of polygons of perimeter <= 20.

Cf. A000228, A057779, A018190, A258019, A258204, A258205, A316193.

(define (A258206 n) (* (/ 1 2) (+ (A258204 n) (A258205 n))))

a(n) = (1/2) * (A258204(n) + A258205(n)).
Other observations. For all n >= 1:
a(n) <= A057779(n).
a(n) <= A258019(n).

a(14)-a(15) from Luca Petrone, Jan 08 2016
a(16)-a(23) from Cyvin, Brunvoll & Cyvin added by Andrey Zabolotskiy, Mar 01 2023
a(24)-a(32) from Bert Dobbelaere, May 12 2025

cp's OEIS Frontend

A001998 Bending a piece of wire of length n+1; walks of length n+1 on a tetrahedron; also non-branched catafusenes with n+2 condensed hexagons.

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

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A197465 Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

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A343577 Number of generalized polyforms on the truncated square tiling with n cells.

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A197159 Number of free poly-[3^4.6]-tiles (holes allowed) with n cells.

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A197459 Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

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A197462 Number of free poly-[4.6.12]-tiles (holes allowed) with n cells (division into triangles is significant).

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A197156 Number of free poly-[3^3.4^2]-tiles (polyhouses) (holes allowed) with n cells.

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A309159 Number of generalized polyforms on the snub square tiling with n cells.

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