A258206 -id:A258206 - cp's OEIS frontend

A267011 Duplicate of A258206.

0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 312, 744, 2291

Offset: 1

dead

A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490, 143552, 683101, 3274826, 15796897, 76581875, 372868101, 1822236628, 8934910362, 43939164263, 216651036012, 1070793308942, 5303855973849, 26323064063884, 130878392115834, 651812979669234, 3251215493161062, 16240020734253127, 81227147768301723, 406770970805865187, 2039375198751047333

Offset: 1

N. J. A. Sloane

From Markus Voege, Nov 24 2009: (Start)
On the difference between this sequence and A038147:
The first term that differs is for n=6; for all subsequent terms, the number of polyhexes is larger than the number of planar polyhexes.
If I recall correctly, polyhexes are clusters of regular hexagons that are joined at the edges and are LOCALLY embeddable in the hexagonal lattice.
"Planar polyhexes" are polyhexes that are GLOBALLY embeddable in the honeycomb lattice.
Example: (Planar) polyhex with 6 cells (x) and a hole (O):
.. x x
. x O x
.. x x
Polyhex with 6 cells that is cut open (I):
.. xIx
. x O x
.. x x
This polyhex is not globally embeddable in the honeycomb lattice, since adjacent cells of the lattice must be joined. But it can be embedded locally everywhere. It is a start of a spiral. For n>6 the spiral can be continued so that the cells overlap.
Illegal configuration with cut (I):
.. xIx
. x x x
.. x x
This configuration is NOT a polyhex since the vertex at
.. xIx
... x
is not embeddable in the honeycomb lattice.
One has to keep in mind that these definitions are inspired by chemistry. Hence, potential molecules are often the motivation for these definitions. Think of benzene rings that are fused at a C-C bond.
The (planar) polyhexes are "free" configurations, in contrast to "fixed" configurations as in A001207 = Number of fixed hexagonal polyominoes with n cells.
A000228 (planar polyhexes) and A001207 (fixed hexagonal polyominoes) differ only by the attribute "free" vs. "fixed," that is, whether the different orientations and reflections of an embedding in the lattice are counted.
The configuration
. x x .... x
.. x .... x x
is counted once as free and twice as fixed configurations.
Since most configurations have no symmetry, (A001207 / A000228) -> 12 for n -> infinity. (End)

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.
A. T. Balaban and Paul von R. Schleyer, "Graph theoretical enumeration of polymantanes", Tetrahedron, (1978), vol. 34, 3599-3609
M. Gardner, Polyhexes and Polyaboloes. Ch. 11 in Mathematical Magic Show. New York: Vintage, pp. 146-159, 1978.
M. Gardner, Tiling with Polyominoes, Polyiamonds and Polyhexes. Chap. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.
J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
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).

John Mason and Robert A. Russell, Table of n, a(n) for n = 1..36
Frédéric Chyzak, Ivan Gutman, and Peter Paule, Predicting the number of hexagonal systems with 24 and 25 hexagons, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.
A. Clarke, Polyhexes
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
M. Keller, Counting polyforms
D. A. Klarner, Cell growth problems, Canad. J. Math. 19 (1967) 851-863.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.
John Mason, Counting polyhexes of size 36, updated Oct 27 2023.
Joseph Myers, Polyomino, polyhex and polyiamond tiling
Ed Pegg, Jr., Illustrations of polyforms
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
N. J. A. Sloane, Illustration of initial terms
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Eric Weisstein's World of Mathematics, Polyhex.

Equals (A006535 + A030225)/2.
Cf. A036359, A002216, A005963, A000228, A001998, A018190.
Cf. A001207, A057973, A364306.
Cf. also A000105, A000577, A103465, A057779, A258206, A258019.

a(13) from Achim Flammenkamp, Feb 15 1999
a(14) from Brendan Owen, Dec 31 2001
a(15) from Joseph Myers, May 05 2002
a(16)-a(20) from Joseph Myers, Sep 21 2002
a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(22)-a(30) from John Mason, Jul 18 2023

A284869 Number of n-step 2-dimensional closed self-avoiding paths on triangular lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 5, 16, 37, 120, 344, 1175, 3807, 13224, 45645, 161705, 575325, 2074088, 7521818, 27502445, 101134999, 374128188

Offset: 1

Luca Petrone, Apr 04 2017

Differs from A057729 beginning at n = 11, since that sequence includes triangular polyominoes with holes.

a(n) is the number of simply connected polyiamonds with perimeter n. - Walter Trump, Nov 29 2023

Rade Doroslovački, Ivan Stojmenović and Ratko Tošić, Generating and counting triangular systems, BIT Numerical Mathematics, 27 (1987), 18-24. See Table 1.
Hugo Pfoertner, Illustration of ratio A036418(n)/a(n) using Plot2.
Hugo Pfoertner, Illustration of polygons of perimeter <= 11.
Walter Trump, Self-avoiding closed walks on a triangular lattice

Approaches (1/12)*A036418 for increasing n.

Cf. A057729, A258206, A266549, A316196.

a(15) from Hugo Pfoertner, Jun 27 2018

a(16)-a(22) from Walter Trump, Nov 29 2023

A346123 Numbers m such that no self-avoiding walk of length m + 1 on the honeycomb net fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 2, 6, 7, 10, 12, 13, 14, 15, 16, 23, 24, 25, 27, 28, 30, 33, 36, 37, 38, 42, 43, 46, 53, 54, 55, 56, 58, 59, 62

Offset: 1

Hugo Pfoertner, Jul 05 2021

The segments of the walk can make relative turns of +- 60 degrees. The walks may be open or closed.

			Illustration of initial terms:
                               %%% %%% %%%
                           %                %
                         %                    %
      %  %              %                     /%
   %        %          %      a(2) = 2       /  %
  %__________%        %                     /    %
  %   L = 1  %       %                     /      %
   %  D = 1 %        %   L = 2, D = 1.732 /       %
      %  %           %                   /        %
                      %                 / Pi/3   %
    a(1) = 1           %-------------- .  .  . .%
                        %                      %
                          %                  %
                              %%% %%% %%%
.
           %%% %%%% %%%                         %%% %%%% %%%
        %                %                   %                %
      %                    %               %                  \ %
     %                      %             %                    \ %
    %                        %           %                      \ %
   %                          %         %                        \ %
  %                            %       %                          \ %
  %.      L = 3, D = 2.00     .%       %.      L = 4, D = 2.00     .%
  % \                        / %       % \                        / %
   % \                      / %         % \                      / %
    % \                    / %           % \                    / %
     % \                  / %             % \                  / %
       % ---------------- %                 % ---------------- %
           %%% %%% %%%                          %%% %%% %%%
.
            %%% %%% %%%                          %%% %%% %%%
        % ______________ %                   % ______________ %
      %                  \ %               % /                \ %
     %                    \ %             % /                  \ %
    %                      \ %           % /                    \ %
   %                        \ %         % /       a(3) = 6       \ %
  %                          \ %       % /                        \ %
  %.      L = 5, D = 2.00     .%       %.      L = 6, D = 2.00     .%
  % \                        / %       % \                        / %
   % \                      / %         % \                      / %
    % \                    / %           % \                    / %
     % \                  / %             % \                  / %
       % ---------------- %                 % ---------------- %
           %%% %%%% %%%                         %%% %%%% %%%
.
The path of minimum diameter of length 7 requires an enclosing circle of D = 3.055, which is greater than the previous minimum diameter of D = 2.00 corresponding to a(3) = 6. No path of length 8 exists that fits into a circle of D = 3.055, thus a(4) = 7.
See link for illustrations of terms corresponding to diameters D <= 9.85.

Hugo Pfoertner, Examples of paths of maximum length.

Cf. A122223, A127399, A127400, A127401, A258206, A266925.

Cf. A346124-A346132 similar to this sequence with other sets of turning angles.

a(n+1) >= a(n) + 1 for n > 1; a(1) = 1.

A316195 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/5, +-3Pi/5, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 2, 1, 18, 45, 441

Offset: 1

Hugo Pfoertner, Jun 26 2018

Holes are excluded, i.e., the boundary path may nowhere touch or intersect itself.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of polygons of perimeter <= 14.

Cf. A306175, A258206, A266549, A284869.

A057779 Number of hexagonal polyominoes (or polyhexes, A000228) with perimeter 2n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 98, 313, 750, 2308, 6270

Offset: 1

N. J. A. Sloane, Oct 29 2000

Cf. A000228, A000105, A057730, A258206 (counts only polyhexes without holes).

Link updated by William Rex Marshall, Dec 16 2009

a(13)-a(16) from John Mason, Jul 26 2021

A316197 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/7, +-3Pi/7, +-5*Pi/7, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 3, 4, 83, 533, 8329

Offset: 1

Hugo Pfoertner, Jun 28 2018

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of polygons of perimeter <= 12.

Cf. A306177, A258206, A266549, A284869, A316195.

A258204 Number of one-sided strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotation.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 16, 23, 80, 183, 563

Offset: 1

Antti Karttunen, May 31 2015

For n >= 1, a(n) gives the total number of terms k in A258003 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

Cf. A057779, A258003, A258017, A258205, A258206.

(define (A258204 n) (let loop ((k (+ 1 (expt 2 (+ n n)))) (c 0)) (cond ((pow2? k) c) (else (loop (+ 1 k) (+ c (if (isA258003? k) 1 0)))))))
(define (pow2? n) (let loop ((n n) (i 0)) (cond ((zero? n) #f) ((odd? n) (and (= 1 n) i)) (else (loop (/ n 2) (1+ i)))))) ;; Gives non-false only when n is a power of two.
;; Code for isA258003? given in A258003.

Other identities and observations. For all n >= 1:
a(n) = 2*A258206(n) - A258205(n).
a(n) <= A258017(n).

A258019 Number of fusenes (not necessarily planar) 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, 313

Offset: 1

Antti Karttunen, Jun 02 2015

A fusene is a benzenoid (a polyhex) which has a single component of boundary edges (that is, no holes). Including also geometrically nonplanar configurations allows helicene-like self-touching or self-overlapping structures. Thus this sequence differs from A258206 for the first time at n=13 as here a(13) = 313 [while A258206(13) = 312] because the smallest such nonplanar structure is 26-edge [6]Helicene, which is encoded by one-capped binary code 131821024 (= A258013(875) = A258015(113)). Please see the illustrations at the Wikipedia page. Note that although in their three-dimensional conformation molecules like [6]Helicene and other [n]Helicenes with n >= 6 have two different chiralities (resulting from the handedness of the helicity itself), in this count of abstract combinatorial objects they are considered achiral because of their bilateral symmetry.

If one counts these structures by the number of hexes (instead of perimeter length), one obtains sequence 1, 1, 3, 7, 22, 82, ... (probably A108070).

Guo, Hansen, Zheng, Boundary uniqueness of fusenes, Discrete Applied Mathematics 118 (2002), pp. 209-222.
Eric Weisstein's World of Mathematics, Fusene
Wikipedia, Helicene

Cf. A108070, A258017, A258018, A258206.

(define (A258019 n) (* (/ 1 2) (+ (A258017 n) (A258018 n))))

a(n) = (1/2) * (A258017(n) + A258018(n)). [1/2 times the count of one-sided fusenes + the count of fusenes with bilateral symmetry (subset of the former)].
Other observations:
For all n, a(n) >= A258206(n).

A258205 Number of strictly non-overlapping holeless polyhexes of perimeter 2n with bilateral symmetry, counted up to rotation.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 1, 8, 5, 20, 11, 61

Offset: 1

Antti Karttunen, May 31 2015

This sequence counts by perimeter length those holeless polyhexes that stay same when they are flipped over and rotated appropriately.

For n >= 1, a(n) gives the total number of terms k in A258005 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

Cf. A057779, A258005, A258018, A258204, A258206.

(define (A258205 n) (let loop ((k (+ 1 (expt 2 (+ n n)))) (c 0)) (cond ((pow2? k) c) (else (loop (+ 1 k) (+ c (if (isA258005? k) 1 0)))))))
(define (pow2? n) (let loop ((n n) (i 0)) (cond ((zero? n) #f) ((odd? n) (and (= 1 n) i)) (else (loop (/ n 2) (1+ i)))))) ;; Gives non-false only when n is a power of two.
;; Code for isA258005? given in A258005.

Other identities and observations. For all n >= 1:
a(n) = 2*A258206(n) - A258204(n).
a(n) <= A258018(n).

cp's OEIS Frontend

A267011 Duplicate of A258206.

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A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

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A284869 Number of n-step 2-dimensional closed self-avoiding paths on triangular lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

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A346123 Numbers m such that no self-avoiding walk of length m + 1 on the honeycomb net fits into the smallest circle that can enclose a walk of length m.

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A316195 Number of self-avoiding polygons with perimeter 2*n and sides = 1 that have vertex angles from the set +-Pi/5, +-3*Pi/5, not counting rotations and reflections as distinct.

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A057779 Number of hexagonal polyominoes (or polyhexes, A000228) with perimeter 2n.

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A316197 Number of self-avoiding polygons with perimeter 2*n and sides = 1 that have vertex angles from the set +-Pi/7, +-3*Pi/7, +-5*Pi/7, not counting rotations and reflections as distinct.

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A258204 Number of one-sided strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotation.

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A258019 Number of fusenes (not necessarily planar) of perimeter 2n, counted up to rotations and turning over.

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A258205 Number of strictly non-overlapping holeless polyhexes of perimeter 2n with bilateral symmetry, counted up to rotation.

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A316195 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/5, +-3Pi/5, not counting rotations and reflections as distinct.

A316197 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/7, +-3Pi/7, +-5*Pi/7, not counting rotations and reflections as distinct.