A001683 -id:A001683 - cp's OEIS frontend

A295633 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation.

1, 1, 1, 1, 1, 1, 1, 2, 4, 4, 1, 2, 8, 12, 6, 1, 3, 16, 40, 43, 19, 1, 3, 25, 93, 165, 143, 49, 1, 4, 40, 197, 505, 712, 504, 150, 1, 4, 56, 364, 1274, 2548, 2912, 1768, 442, 1, 5, 80, 646, 2878, 7672, 12400, 11976, 6310, 1424, 1, 5, 105, 1050, 5880, 19992, 42840, 58140, 48450, 22610, 4522

Offset: 3

Andrew Howroyd, Nov 24 2017

			Triangle begins: (n >= 3, k >= 1)
1;
1, 1;
1, 1,  1;
1, 2,  4,   4;
1, 2,  8,  12,    6;
1, 3, 16,  40,   43,   19;
1, 3, 25,  93,  165,  143,   49;
1, 4, 40, 197,  505,  712,  504,  150;
1, 4, 56, 364, 1274, 2548, 2912, 1768, 442;
...

Andrew Howroyd, Table of n, a(n) for n = 3..1277

Row sums are A003455.
Column k=3 is A003451.
Diagonals include A001683, A220881, A003445, A220882.
Cf. A295224, A295495, A295634.

\\ See A295495 for DissectionsModCyclic()
T=DissectionsModCyclic(apply(i->y, [1..12]));
for(n=3, #T, for(k=1, n-2, print1(polcoeff(T[n], k), ", ")); print)

A006342 Coloring a circuit with 4 colors.

Original entry on oeis.org

1, 1, 4, 10, 31, 91, 274, 820, 2461, 7381, 22144, 66430, 199291, 597871, 1793614, 5380840, 16142521, 48427561, 145282684, 435848050, 1307544151, 3922632451, 11767897354, 35303692060, 105911076181, 317733228541, 953199685624, 2859599056870, 8578797170611

Offset: 0

N. J. A. Sloane, Simon Plouffe

Also equal to the number of set partitions of {1,2,...,n+2} with at most 4 parts such that each part does not contain both i,i+1 for 1<=iMike Zabrocki, Sep 08 2020

Also a(n) equals the number of color-complete multipoles with n terminals (that is, having all the states allowed by the Parity Lemma). - Miquel A. Fiol, May 27 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. R. Bernhart, Topics in Graph Theory Related to the Five Color Conjecture, Ph.D. Dissertation, Kansas State Univ., 1974.
F. R. Bernhart, Fundamental chromatic numbers, Unpublished. (Annotated scanned copy)
F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977
G. D. Birkhoff, D. C. Lewis, Chromatic polynomials, Trans. Amer. Math. Soc. 60, (1946). 355-451.
Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, arXiv preprint arXiv:1104.4323 [hep-th], 2011.
M. A. Fiol and J. Vilaltella, Some results on the structure of multipoles in the study of snarks, Electron. J. Combin. 22(1) (2015), #P1.45.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

A214142, A214167

[3*3^n/8+1/4+3*(-1)^n/8: n in [0..30]]; // Vincenzo Librandi, Aug 20 2011

A006342:=-(-1+2*z)/(z-1)/(3*z-1)/(z+1); # conjectured by Simon Plouffe in his 1992 dissertation
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]-1 od: seq(a[n], n=1..26); # Zerinvary Lajos, Apr 28 2008

CoefficientList[Series[(1-2 x)/((1-x^2) (1-3 x)),{x,0,30}],x] (* or *) LinearRecurrence[{3,1,-3},{1,1,4},30] (* Harvey P. Dale, Aug 16 2016 *)

Vec((1 - 2*x) / ((1 - x)*(1 + x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 07 2017

G.f.: (1 - 2 x ) / (( 1 - x^2 ) ( 1 - 3 x )).
Binomial transform of A002001 (with interpolated zeros). Partial sums of A054878. E.g.f.: exp(x)(3*cosh(2*x) + 1)/4; a(n) = 3*3^n/8 + 1/4 + 3(-1)^n/8 = Sum_{k=0..n} (3^k + 3(-1)^k)/4. - Paul Barry, Sep 03 2003
a(n) = 2*a(n-1) + 3*a(n-2) - 1, n > 1. - Gary Detlefs, Jun 21 2010
a(n) = a(n-1) + A054878(n-2). - Yuchun Ji, Sep 12 2017
From Colin Barker, Nov 07 2017: (Start)
a(n) = (3^(n+1) + 5) / 8 for n even.
a(n) = (3^(n+1) - 1) / 8 for n odd.
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n > 2.
(End)
a(n) = 3*a(n-1) + (3*(-1)^n - 1)/2 for n > 0. - Yuchun Ji, Dec 05 2019

A221184 Number of colored quivers in the 4-mutation class of a quiver of Dynkin type A_n.

Original entry on oeis.org

1, 1, 3, 19, 118, 931, 7756, 68685, 630465, 5966610, 57805410, 571178751, 5737638778, 58455577800, 602859152496, 6283968796705, 66119469155523, 701526880303315, 7498841128986109, 80696081185766970, 873654669882574580

Offset: 0

N. J. A. Sloane, Jan 22 2013

Also, number of nonequivalent dissections of a polygon into n+1 hexagons by nonintersecting diagonals up to rotation. - Andrew Howroyd, Nov 20 2017

Number of oriented polyominoes composed of n+1 hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Jan 23 2024

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, arXiv:1004.4512 [math.RT], 2010.
Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, J. Algebra and Applications, 12 (2013), #1250133.

Column k=6 of A295224.

Polyominoes: A004127 (unoriented), A369473 (chiral), A143546 (achiral), A001683(n+2) {3,oo}, A005034 {4,oo}, A005038 {5,oo}.

u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
T[n_, k_] := u[n, k, 1] + (If[EvenQ[n], u[n/2, k, 1], 0] - u[n, k, 2])/2 + DivisorSum[GCD[n - 1, k], EulerPhi[#]*u[(n - 1)/#, k, k/#] &]/k;
a[n_] := T[n + 1, 6];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)
p=6; Table[Binomial[(p-1)n,n]/(((p-2)n+1)((p-2)n+2))+If[OddQ[n],0,Binomial[(p-1)n/2,n/2]/((p-2)n+2)]+DivisorSum[GCD[p,n-1],EulerPhi[#]Binomial[((p-1)n+1)/#,(n-1)/#]/((p-1)n+1)&,#>1&],{n,30}] (* Robert A. Russell, Jan 23 2024 *)

a(n) ~ 5^(5*n + 11/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 27/2)). - Vaclav Kotesovec, Jun 15 2018

a(n-1) = A004127(n) + A369473(n) = 2*A004127(n) - A143546(n) = 2*A369473(n) + A143546(n). - Robert A. Russell, Jan 23 2024

a(0)=1 and a(18)-a(20) corrected by Andrew Howroyd, Nov 20 2017

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

Original entry on oeis.org

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).

A038775 a(n) is the number of cycles of the permutation that converts forest(n) of depth-first planar planted binary trees into breadth-first representation.

Original entry on oeis.org

1, 2, 3, 6, 10, 12, 17, 26, 34, 50, 56, 68, 82, 94, 113

Offset: 1

Wouter Meeussen, May 04 2000

The first a(n) terms of A038774 add up to Catalan(n) = A000108(n).

			a(5)=10 since there are 10 cycles in this permutation of forest(5), with lengths 1, 1, 3, 4, 3, 2, 16, 8, 2, 2 summing up to 42=Catalan(5).

Sean A. Irvine, Java program (github)

Cf. A000108, A038774.

Similarly generated sequences: A001683, A002995, A003239, A057507, A057513.

a(13)-a(15) from Sean A. Irvine, May 22 2022

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

A380361 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of Halin graphs on n unlabeled nodes with circuit rank k up to orientation-preserving homeomorphisms, 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, 4, 2, 1, 0, 0, 0, 4, 8, 3, 1, 0, 0, 0, 0, 12, 16, 3, 1, 0, 0, 0, 0, 6, 40, 25, 4, 1, 0, 0, 0, 0, 0, 43, 93, 40, 4, 1, 0, 0, 0, 0, 0, 19, 165, 197, 56, 5, 1, 0, 0, 0, 0, 0, 0, 143, 505, 364, 80, 5, 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 rotation.

			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, 4,  2,  1;
  10 | 0, 0, 0, 4,  8,  3,   1;
  11 | 0, 0, 0, 0, 12, 16,   3,   1;
  12 | 0, 0, 0, 0,  6, 40,  25,   4,  1;
  13 | 0, 0, 0, 0,  0, 43,  93,  40,  4,  1;
  14 | 0, 0, 0, 0,  0, 19, 165, 197, 56,  5,  1;
  ...

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

Row sums are A380360.
Column sums are A003455.
Main diagonal is A000012.
Central coefficients are A001683.
Cf. A295633, A380362.

\\ See PARI Link in A380362 for program code.
{ my(T=A380361rows(12)); for(i=1, #T, print(T[i])) }

T(n,k) = A295633(k, n-k).

A085167 Permutation of natural numbers induced by the Catalan bijection gma085167 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 5, 8, 7, 4, 6, 10, 13, 15, 18, 22, 12, 21, 17, 9, 11, 20, 14, 16, 19, 24, 27, 29, 32, 36, 38, 41, 43, 46, 50, 52, 55, 59, 64, 26, 35, 40, 49, 63, 31, 58, 45, 23, 25, 48, 28, 30, 33, 34, 62, 54, 37, 39, 57, 42, 44, 47, 61, 51, 53, 56, 60, 66, 69, 71, 74, 78

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

Antti Karttunen, Gatomorphisms (With the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse: A085168. a(n) = A085159(A069770(n)). Occurs in A073200. Cf. also A074679, A074680, A085203.

Number of cycles in range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right: A001683. [To be checked.].

A111113 a(2^m) = 1, a(2^m+1) = -1 (m>0), otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 1, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Oct 15 2005

sign

			G.f. = x^2 - x^3 + x^4 - x^5 + x^8 - x^9 + x^16 - x^17 + x^32 - x^33 + ...

A. G. Postnikov, Tauberian Theory and Its Applications, Proc. Steklov. Inst. Math., 144 (1979), translated as Issue 2, 1980. See p. 29.
A. Renyi, On a Tauberian theorem of O. Szasz, Acta Univ. Szeged. Sect. Sci. Math., 11 (1946), 119-123.

{a(n) = if( n<2, 0, [1, -1, 0] [1 + min(2, n - 2^(length(binary(n)) - 1))] )} /* Michael Somos, Aug 03 2009 */

{a(n) = if( n<2, 0, if( n%2, -a(n - 1), n == 2^valuation(n, 2)))} /* Michael Somos, Aug 03 2009 */

Euler transform of A079559 is sequence offset -1. - Michael Somos, Aug 03 2009
G.f.: (1 - x) * (Sum_{k>0} x^(2^k)). - Michael Somos, Aug 03 2009
|a(n)| = A001683(n)(mod 2) for n > 1. - John M. Campbell, Apr 01 2018

A274326 Number of distinct irregular n-gon flexagons.

Original entry on oeis.org

0, 0, 1, 2, 11, 40

Offset: 2

Christian Schroeder, Jun 18 2016

			a(6) = 11, because there are 11 distinct irregular hexaflexagons.

Les Pook, Flexagons inside out, Cambridge University Press, 2003, p. 84.

Cf. A001683.

cp's OEIS Frontend

A295633 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation.

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A006342 Coloring a circuit with 4 colors.

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A221184 Number of colored quivers in the 4-mutation class of a quiver of Dynkin type A_n.

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

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A038775 a(n) is the number of cycles of the permutation that converts forest(n) of depth-first planar planted binary trees into breadth-first representation.

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

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A380361 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of Halin graphs on n unlabeled nodes with circuit rank k up to orientation-preserving homeomorphisms, 3 <= k <= n-1.

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A085167 Permutation of natural numbers induced by the Catalan bijection gma085167 acting on symbolless S-expressions encoded by A014486/A063171.

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A111113 a(2^m) = 1, a(2^m+1) = -1 (m>0), otherwise a(n) = 0.