The OEIS is supported by the many generous donors to the OEIS Foundation.



Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000309 Number of rooted planar bridgeless cubic maps with 2n nodes.
(Formerly M3601 N1460)
1, 1, 4, 24, 176, 1456, 13056, 124032, 1230592, 12629760, 133186560, 1436098560, 15774990336, 176028860416, 1990947110912, 22783499599872, 263411369705472, 3073132646563840, 36143187370967040, 428157758086840320, 5105072641718353920, 61228492804372561920 (list; graph; refs; listen; history; text; internal format)



Also counts rooted planar non-separable triangulations with 3n edges. - Valery A. Liskovets, Dec 01 2003

Equivalently, rooted planar loopless triangulations with 2n triangles. - Noam Zeilberger, Oct 06 2016

Description trees of type (2,2) with n edges. (A description tree of type (a,b) is a rooted plane tree where every internal node is labeled by an integer between a and [b + sum of labels of its children], every leaf is labeled a, and the root is labeled [b + sum of labels of its children]. See Definition 1 and Section 5.2 of Cori and Schaeffer 2003.) - Noam Zeilberger, Oct 08 2017

The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018


C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.

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 = 0..100

Marie Albenque, Dominique Poulalhon, A Generic Method for Bijections between Blossoming Trees and Planar Maps, Electron. J. Combin., 22 (2015), #P2.38.

Dario Benedetti, Sylvain Carrozza, Reiko Toriumi, Guillaume Valette, Multiple scaling limits of U(N)^2 X O(D) multi-matrix models, arXiv:2003.02100 [math-ph], 2020.

Olivier Bernardi, Bijective counting of Kreweras walks and loopless triangulations, Journal of Combinatorial Theory, Series A 114:5 (2007), 931-956.

Junliang Cai, Yanpei Liu, The enumeration of rooted nonseparable nearly cubic maps, Discrete Math. 207 (1999), no. 1-3, 9--24. MR1710479 (2000g:05074). See (31).

Robert Cori and Gilles Schaeffer, Description trees and Tutte formulas, Theoretical Computer Science 292:1 (2003), 165-183.

S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)

Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.

R. C. Mullin, On counting rooted triangular maps, Canad. J. Math., v.17 (1965), 373-382.

Elena Patyukova, Taylor Rottreau, Robert Evans, Paul D. Topham, Martin J. Greenall, Hydrogen Bonding Aggregation in Acrylamide: Theory and Experiment, Macromolecules (2018) Vol. 51, No. 18, 7032-7043. Also arXiv:1805.09878 [math.CA], 2018.

W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.

W. T. Tutte, On the enumeration of four-colored maps, SIAM J. Appl. Math., 17 (1969), 454-460.

Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.

Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv:1803.10080 [math.LO], March 2018 (A revised version of a 2017 conference paper)

Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Part 2, Rutgers Experimental Math Seminar, Sep 13 2018.

Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.


a(n) = 2^(n-1) * A000139(n) for n > 0.

a(n) = 4*a(n-1)*binomial(3*n, 3) / binomial(2*n+2, 3).

a(n) = 2^n*(3*n)!/ ( (n+1)!*(2*n+1)! ).

G.f.: (1/(6*x)) * (hypergeom([ -2/3, -1/3],[1/2],(27/2)*x)-1). - Mark van Hoeij, Nov 02 2009

a(n) ~ 3^(3*n+1/2)/(sqrt(Pi)*2^(n+2)*n^(5/2)). - Ilya Gutkovskiy, Oct 06 2016

D-finite with recurrence (n+1)*(2*n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Nov 02 2018

a(n) = -(-2)^(n-1)*(3*n+2)*hypergeom([-3*(n+1),-n,-n+1/3], [-n-1,-n-2/3], 1). The a(n) are values of the polynomials A358091. - Peter Luschny, Oct 28 2022


a := n -> 2^(n+1)*(3*n)!/(n!*(2*n+2)!);

A000309 := n -> -(-2)^(n-1)*(3*n+2)*hypergeom([-3*(n+1), -n, -n+1/3], [-n-1, -n-2/3], 1): seq(simplify(A000309(n)), n = 0..21); # Peter Luschny, Oct 28 2022


f[n_] := 2^n(3n)!/((n + 1)!(2n + 1)!); Table[f[n], {n, 0, 19}] (* Robert G. Wilson v, Sep 21 2004 *)

Join[{1}, RecurrenceTable[{a[1]==1, a[n]==4a[n-1] Binomial[3n, 3]/ Binomial[2n+2, 3]}, a[n], {n, 20}]] (* Harvey P. Dale, May 11 2011 *)


(PARI) a(n) = 2^(n+1)*(3*n)!/(n!*(2*n+2)!); \\ Michel Marcus, Aug 09 2014

(Magma) [2^(n+1)*Factorial(3*n)/(Factorial(n)*Factorial(2*n+2)): n in [0..20]]; // Vincenzo Librandi, Aug 10 2014

(Sage) [2^n*factorial(3*n)/(factorial(n+1)*factorial(2*n+1))for n in range(20)] # G. C. Greubel Nov 29 2018

(GAP) List([0..20], n -> 2^(n+1)*Factorial(3*n)/(Factorial(n)* Factorial(2*n+2))); # G. C. Greubel, Nov 29 2018


Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.

Cf. A000139, A000264, A000356, A002005, A006335, A358091.

Sequence in context: A215709 A103334 A156017 * A112914 A308543 A007846

Adjacent sequences: A000306 A000307 A000308 * A000310 A000311 A000312




N. J. A. Sloane and Robert G. Wilson v


Definition clarified by Michael Albert, Oct 24 2008



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 12:40 EST 2022. Contains 358441 sequences. (Running on oeis4.)