A000207 -id:A000207 - cp's OEIS frontend

A369314 Number of chiral pairs of polyominoes composed of n triangular cells of the hyperbolic regular tiling with Schläfli symbol {3,oo}.

1, 2, 7, 22, 68, 214, 691, 2240, 7396, 24702, 83469, 284928, 981814, 3410990, 11939752, 42075308, 149180356, 531866972, 1905872189, 6861162880, 24805796984, 90035940942, 327988261992, 1198853954688, 4395798528850

Offset: 4

Robert A. Russell, Jan 19 2024

A stereographic projection of the {3,oo} tiling on the Poincaré disk can be obtained via the Christensson link. Each member of a chiral pair is a reflection but not a rotation of the other.

			________      ________   ________      ________   ________      ________
\  /\  /\    /\  /\  /   \  /\  /\    /\  /\  /   \  /\  /\    /\  /\  /
 \/__\/__\  /__\/__\/     \/__\/__\  /__\/__\/     \/__\/__\  /__\/__\/
                           \  /          \  /           \  /  \  /
a(4)=1; a(5)=2.             \/            \/             \/    \/

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Polyominoes: A001683(n+2) (oriented), A000207 (unoriented), A208355(n-1) (achiral).

Table[Binomial[2n,n]/(2(n+1)(n+2))-If[OddQ[n],Binomial[n,(n+1)/2]/n,Binomial[n,n/2]/(n+2)]/2+If[Divisible[n-1,3],Binomial[(2n+1)/3,(n-1)/3]/(2n+1),0],{n,4,20}]

a(n) = C(2n,2)/(2(n+1)(n+2)) - [2\(n+1)]*C(n,(n+1)/2)/(2n) - [2\n]*C(n,n/2)/(2n+4) + [3\(n-1)]*C((2n+1)/3,(n-1)/3)/(2n+1).

a(n) = A001683(n+2) - A000207(n) = (A001683(n+2) - A208355(n-1)) / 2 = A000207(n) - A208355(n-1).

A139434 Frieze pattern with 4 rows, read by diagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2

Offset: 0

N. J. A. Sloane, Jun 09 2008

Period 20: repeat [1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1]. - Wesley Ivan Hurt, Jun 05 2016

Every frieze is defined by its quiddity (the row below the row of 1's), which corresponds to the counts of triangles at vertices of a dissection of a regular polygon. The quiddity of this frieze is A135352. One can say that this frieze pattern has width 2 (not counting the rows of 1's), 4, or 5 (implying the additional row of 0's; this is also the period of the pattern and the number of vertices in the dissected polygon), depending on the convention. In any case, friezes of given width are enumerated by A000207 if we identify shifts and mirror images, otherwise by A000108. A000207(3) = 1 means that this is the only frieze of this width, and it has A000108(3) = 5 different horizontal shifts or reflections. The A000207(4) = 3 friezes having width 1 greater than this one are A139438, A139458, and one more with quiddity 1, 3, 1, 3, 1, 3, ... (currently not in the OEIS). The only frieze having width 1 less than this one has quiddity 1, 2, 1, 2, ... (A245477 can be interpreted as representing that frieze pattern). - Andrey Zabolotskiy, Jan 30 2024

			The frieze pattern is
...1 1 1 1 1 1 1 ...
....1 2 2 1 3 1 2 ...
.....1 3 1 2 2 1 3 ...
......1 1 1 1 1 1 1 ...

J. H. Conway and R. K. Guy, The Book of Numbers. New York: Springer-Verlag, p. 97, 1996.

Cf. A139438, A139458, A135352, A000108, A000207, A245477.

A380362 Triangle read by rows: T(n,k) is the number of Halin graphs on n unlabeled nodes with circuit rank k, 3 <= k <= n-1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 3, 2, 1, 0, 0, 0, 3, 6, 3, 1, 0, 0, 0, 0, 7, 11, 3, 1, 0, 0, 0, 0, 4, 24, 17, 4, 1, 0, 0, 0, 0, 0, 24, 51, 26, 4, 1, 0, 0, 0, 0, 0, 12, 89, 109, 36, 5, 1, 0, 0, 0, 0, 0, 0, 74, 265, 194, 50, 5, 1, 0, 0, 0, 0, 0, 0, 27, 371, 660, 345, 65, 6, 1

Offset: 4

Andrew Howroyd, Jan 25 2025

The circuit rank is equal to the number of leaves on the tree before it is extended into a Halin graph by joining up the leaves.
The main diagonal of the graph corresponds with the wheel graphs which have the greatest circuit rank of all Halin graphs.
T(n,k) is also the number of nonequivalent dissections of a k-gon into n-k polygons by nonintersecting diagonals up to rotations and reflections.

			Triangle begins:
  n\k| 3  4  5  6  7   8   9   10  11  12  13
-----+----------------------------------------
   4 | 1;
   5 | 0, 1;
   6 | 0, 1, 1;
   7 | 0, 0, 1, 1;
   8 | 0, 0, 1, 2, 1;
   9 | 0, 0, 0, 3, 2,  1;
  10 | 0, 0, 0, 3, 6,  3,  1;
  11 | 0, 0, 0, 0, 7, 11,  3,   1;
  12 | 0, 0, 0, 0, 4, 24, 17,   4,  1;
  13 | 0, 0, 0, 0, 0, 24, 51,  26,  4,  1;
  14 | 0, 0, 0, 0, 0, 12, 89, 109, 36,  5,  1;
   ...

Andrew Howroyd, Table of n, a(n) for n = 4..1278 (first 50 rows)
Andrew Howroyd, Formulas and PARI Program, Jan 2025.
Eric Weisstein's World of Mathematics, Halin Graph.
Wikipedia, Circuit rank.
Wikipedia, Halin graph.

Row sums are A346779.
Column sums are A001004.
Main diagonal is A000012.
Central coefficients are A000207.
Cf. A295634, A380361.

\\ See PARI Link for program code.
{ my(T=A380361rows(12)); for(i=1, #T, print(T[i])) }

T(n,k) = A295634(k, n-k).

A007282 Number of hexaflexagons with 3n triangles that can be folded from a straight strip of paper.

Original entry on oeis.org

1, 1, 1, 4, 14, 74, 434, 2876, 19848, 143306, 1062149, 8058223, 62259820, 488630360, 3886211100, 31267852668

Offset: 1

N. J. A. Sloane, frb6006(AT)cs.rit.edu (Frank R. Bernhart)

M. Kosters, A theory of hexaflexagons, Nieuw Archief Wisk., 17 (1999), 349-362.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A001683, A000207.

More terms from Mike Godfrey (m.godfrey(AT)umist.ac.uk), Apr 02 2002

A330659 The number of free polyiamonds with n cells on an order-7 triangular tiling of the hyperbolic plane.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 12, 27, 78, 208, 637, 1870, 5797, 17866, 56237, 177573, 566904, 1818527, 5874180, 19065038

Offset: 0

Peter Kagey, Mar 01 2020

This gives the number of polyforms with n cells in the hyperbolic tiling with Schläfli symbol {3,7}.

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, Order-7 triangular tiling
Wikipedia, Polyiamond

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

a(11)-a(19) from Ed Wynn, Feb 14 2021

A001895 Number of rooted planar 2-trees with n nodes.

Original entry on oeis.org

1, 2, 4, 12, 34, 111, 360, 1226, 4206, 14728, 52024, 185824, 668676, 2424033, 8839632, 32412270, 119410390, 441819444, 1641032536, 6116579352, 22870649308, 85764947502, 322476066224, 1215486756372, 4591838372044

Offset: 1

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.28).
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).

T. D. Noe, Table of n, a(n) for n=1..200
F. Harary and E. M. Palmer, Enumeration of self-converse digraphs, Mathematika, 13 (1966), 151-157.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000108, A000207.

Rest[CoefficientList[Series[(4-8x^2-Sqrt[1-4x]-(3+2x)Sqrt[1-4x^2])/ (8x^2),{x,0,30}],x]] (* Harvey P. Dale, Aug 08 2011 *)

G.f.: (4-8*x^2-sqrt(1-4*x)-(3+2*x)*sqrt(1-4*x^2))/(8*x^2).
a(n) ~ 4^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013
Recurrence: (n+1)*(n+2)*(8*n^3 - 43*n^2 + 67*n - 36)*a(n) = 4*n*(n+1)*(8*n^3 - 39*n^2 + 41*n - 3)*a(n-1) + 4*(8*n^5 - 43*n^4 + 80*n^3 - 26*n^2 - 61*n + 36)*a(n-2) - 8*(n-3)*(2*n-3)*(8*n^3 - 19*n^2 + 5*n - 4)*a(n-3). - Vaclav Kotesovec, Aug 13 2013

More terms from Vladeta Jovovic, Aug 24 2001

A332930 Number of polyominoes with n cells on the order-6 square tiling of the hyperbolic plane.

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 60, 255, 1193, 5862, 29875, 155995, 830539, 4489352, 24581675

Offset: 0

Peter Kagey, Mar 02 2020

The order-6 square tiling has Schläfli symbol {4,6}.

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, Order-6 square tiling
Wikipedia, Polyomino

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

a(9)-a(14) from Ed Wynn, Feb 16 2021

A000953 Number of free nonplanar polyenoids with n nodes.

Original entry on oeis.org

1, 5, 24, 109, 465, 1943, 7827, 31095, 121356, 469235, 1797376, 6844290, 25928036

Offset: 7

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

a(n) = A000207(n) - A000942(n). - Sean A. Irvine, Oct 15 2015

a(16)-a(19) from Sean A. Irvine, Oct 15 2015

A007499 Number of cases considered in a particular algorithm for enumerating hexaflexagrams.

Original entry on oeis.org

2, 3, 4, 12, 20, 55, 127, 371, 1037, 3249, 10071, 32913, 108112, 363618, 1233938, 4244102, 14716095, 51480027, 181312939

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, circa 1976

See O'Reilly reference for precise definition.

Thomas J. O'Reilly, Classifying and Counting Hexaflexagrams, J. Rec. Math., 8 (1976), 182-187.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sean A. Irvine, Java program (github)
Thomas J. O'Reilly, Classifying and Counting Hexaflexagrams, J. Rec. Math., 8 (1976), 182-187. [Annotated scanned copy]

Cf. A000207.

Title clarified and a(11)-a(19) from Sean A. Irvine, Jul 12 2022

A063786 G.f.: A(x) = (x-2x^2-2x^3-(1+x)sqrt(1-4x^2)+sqrt(1-4x^6))/(2x^2).

Original entry on oeis.org

1, 1, 1, 2, 5, 5, 14, 14, 41, 42, 132, 132, 429, 429, 1428, 1430, 4862, 4862, 16796, 16796, 58781, 58786, 208012, 208012, 742900, 742900, 2674426, 2674440, 9694845, 9694845, 35357670, 35357670, 129644748, 129644790, 477638700, 477638700

Offset: 2

Vladeta Jovovic, Aug 24 2001

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.30). ( A(x) = s_1 bar (x) ).

Cf. A000108, A000207.

cp's OEIS Frontend

A369314 Number of chiral pairs of polyominoes composed of n triangular cells of the hyperbolic regular tiling with Schläfli symbol {3,oo}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A139434 Frieze pattern with 4 rows, read by diagonals.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A380362 Triangle read by rows: T(n,k) is the number of Halin graphs on n unlabeled nodes with circuit rank k, 3 <= k <= n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A007282 Number of hexaflexagons with 3n triangles that can be folded from a straight strip of paper.

Views

Author

Keywords

References

Crossrefs

Extensions

A330659 The number of free polyiamonds with n cells on an order-7 triangular tiling of the hyperbolic plane.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A001895 Number of rooted planar 2-trees with n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A332930 Number of polyominoes with n cells on the order-6 square tiling of the hyperbolic plane.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A000953 Number of free nonplanar polyenoids with n nodes.

Views

Author

Keywords

References

Links

Formula

Extensions

A007499 Number of cases considered in a particular algorithm for enumerating hexaflexagrams.

Views

Author

Keywords

Comments

References

Links

A063786 G.f.: A(x) = (x-2x^2-2x^3-(1+x)sqrt(1-4x^2)+sqrt(1-4x^6))/(2x^2).