A108070 -id:A108070 - cp's OEIS frontend

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

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).

A259351 Number of generalized fusenes with n pentagons.

Original entry on oeis.org

1, 1, 2, 4, 7, 18, 35, 87, 206, 527, 1337, 3524, 9262, 24772, 66402, 179589, 487435, 1330708, 3645754, 10031143, 27691894, 76708835, 213122892, 593851624, 1659104995, 4646951873, 13046181167, 36708619646, 103505545079, 292430771955, 827749791118, 2347201206583, 6667102499129, 18968089092814

Offset: 1

Pieter Goetschalckx, Jun 24 2015

nonn

G. Brinkmann, O. Delgado-Friedrichs, S. Lisken, A. Peeters, and N. Van Cleemput, Cage - a virtual environment for studying some special classes of plane graphs
G. Brinkmann and P. Goetschalckx, Enumeration of Generalized Fusenes, MATCH: Communications in Mathematical and in Computer Chemistry, 74.3 (2015), 505-520.
Eric Weisstein's World of Mathematics, Fusene

Cf. A108070, A259352, A259353, A259354, A259355.

A259352 Number of generalized fusenes with n heptagons.

Original entry on oeis.org

1, 1, 3, 10, 44, 249, 1513, 10002, 68455, 482571, 3470782, 25391403, 188353848, 1414150643, 10728851795, 82151290333, 634203702080, 4931994064073, 38607981904156, 304032521364835

Offset: 1

Pieter Goetschalckx, Jun 24 2015

nonn

G. Brinkmann, O. Delgado Friedrichs, S. Lisken, A. Peeters, N. Van Cleemput, CaGe - a Virtual Environment for Studying Some Special Classes of Plane Graphs
G. Brinkmann and P. Goetschalckx, Enumeration of Generalized Fusenes, MATCH: Communications in Mathematical and in Computer Chemistry, 74(3) (2015), 505-520.
Eric Weisstein's World of Mathematics, Fusene

Cf. A108070, A259351, A259353, A259354, A259355.

A259353 Number of generalized fusenes with n octagons.

Original entry on oeis.org

1, 1, 4, 14, 85, 598, 4837, 41570, 372830, 3436289, 32373742, 310407548, 3020894755, 29776585115, 296777846799, 2986882651510, 30321553394791

Offset: 1

Pieter Goetschalckx, Jun 24 2015

G. Brinkmann, O. Delgado Friedrichs, S. Lisken, A. Peeters, N. Van Cleemput, CaGe - a Virtual Environment for Studying Some Special Classes of Plane Graphs
G. Brinkmann and P. Goetschalckx, Enumeration of Generalized Fusenes, MATCH: Communications in Mathematical and in Computer Chemistry, 74(3) (2015), 505-520.
Eric Weisstein's World of Mathematics, Fusene

Cf. A108070, A259351, A259352, A259354, A259355.

A259354 Number of generalized fusenes with n nonagons.

Original entry on oeis.org

1, 1, 4, 19, 136, 1226, 12259, 130435, 1441755, 16390266, 190484723, 2254489731, 27096464528, 329995577356, 4065180800962, 50584233942184

Offset: 1

Pieter Goetschalckx, Jun 24 2015

G. Brinkmann, O. Delgado Friedrichs, S. Lisken, A. Peeters, N. Van Cleemput, CaGe - a Virtual Environment for Studying Some Special Classes of Plane Graphs
G. Brinkmann and P. Goetschalckx, Enumeration of Generalized Fusenes, MATCH: Communications in Mathematical and in Computer Chemistry, 74(3) (2015), 505-520.
Eric Weisstein's World of Mathematics, Fusene

Cf. A108070, A259351, A259352, A259353, A259355.

A259355 Number of generalized fusenes with n decagons.

Original entry on oeis.org

1, 1, 5, 24, 215, 2256, 26925, 339523, 4456349, 60149731, 830492066, 11681979493, 166929456330, 2417719752083, 35429035715975

Offset: 1

Pieter Goetschalckx, Jun 24 2015

G. Brinkmann, O. Delgado Friedrichs, S. Lisken, A. Peeters, N. Van Cleemput, CaGe - a Virtual Environment for Studying Some Special Classes of Plane Graphs
G. Brinkmann and P. Goetschalckx, Enumeration of Generalized Fusenes, MATCH: Communications in Mathematical and in Computer Chemistry, 74(3) (2015), 505-520.
Eric Weisstein's World of Mathematics, Fusene

Cf. A108070, A259351, A259352, A259353, A259354.

A108071 Number of inner dual graphs of planar polyhexes with n hexagons.

Original entry on oeis.org

1, 1, 2, 4, 8, 21, 53, 151, 458, 1477, 4918, 16956, 59494, 212364, 766753, 2796876, 10284793, 38096072, 141998218, 532301941, 2005638293, 7592441954, 28865031086, 110174528925, 422064799013, 1622379252093

Offset: 1

Gunnar Brinkmann, Jun 05 2005

nonn

			For n = 4, the a(n) = 4 graphs are: the 4-path, which is the inner dual of 4 polyhexes out of A018190(4) = 7 (each of the others is an inner dual of a single polyhex); the paw graph; the diamond graph; the claw graph.

Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A constructive enumeration of fusenes and benzenoids, Journal of Algorithms, Volume 45, Issue 2, November 2002, Pages 155-166.
Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons, J. Chem. Inf. Comput. Sci. 2003, 43, 3, 842-851.

Cf. A018190, A108070, A108072.

Name corrected by Andrey Zabolotskiy, Oct 01 2022

A108072 Number of inner dual graphs of planar polyhexes with n hexagons having no nontrivial symmetry.

Original entry on oeis.org

1, 0, 0, 0, 0, 5, 22, 90, 342, 1247, 4491, 16095, 57906, 209170, 760830, 2784913, 10262649, 38051063, 141914613, 532131882, 2005320952, 7591794561, 28863820538, 110172051829, 422060152511, 1622369728951

Offset: 1

Gunnar Brinkmann, Jun 05 2005

nonn

Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A constructive enumeration of fusenes and benzenoids, Journal of Algorithms, Volume 45, Issue 2, November 2002, Pages 155-166.
Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons, J. Chem. Inf. Comput. Sci. 2003, 43, 3, 842-851.

Cf. A018190, A108070, A108071.

Name corrected by Andrey Zabolotskiy, Oct 01 2022

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

cp's OEIS Frontend

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

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A259351 Number of generalized fusenes with n pentagons.

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A259352 Number of generalized fusenes with n heptagons.

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A259353 Number of generalized fusenes with n octagons.

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A259354 Number of generalized fusenes with n nonagons.

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A259355 Number of generalized fusenes with n decagons.

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A108071 Number of inner dual graphs of planar polyhexes with n hexagons.

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A108072 Number of inner dual graphs of planar polyhexes with n hexagons having no nontrivial symmetry.

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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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