A000256 -id:A000256 - cp's OEIS frontend

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

1, 1, 3, 13, 68, 399, 2530, 16965, 118668, 857956, 6369883, 48336171, 373537388, 2931682810, 23317105140, 187606350645, 1524813969276, 12504654858828, 103367824774012, 860593023907540, 7211115497448720, 60776550501588855

Offset: 0

N. J. A. Sloane

Number of rooted loopless planar maps with n edges. E.g., there are a(2)=3 loopless planar maps with 2 edges: two rooted paths (.-.-.) and one digon (.=.). - Valery A. Liskovets, Sep 25 2003
Number of intervals (i.e., ordered pairs (x,y) such that x<=y) in the Tamari lattice (rotation lattice of binary trees) of size n (see Pallo and Chapoton references). - Ralf Stephan, May 08 2007, Jean Pallo (Jean.Pallo(AT)u-bourgogne.fr), Sep 11 2007
Number of rooted triangulations of type [n, 0] (see Brown paper eq (4.8)). - Michel Marcus, Jun 23 2013
Equivalently, number of rooted bridgeless planar maps with n edges. - Noam Zeilberger, Oct 06 2016
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
Number of uniquely sorted permutations of [2n+1] that avoid the pattern 231. Also the number of uniquely sorted permutations of [2n+1] that avoid 132. - Colin Defant, Jun 13 2019
The sequence 1,3,13,68,... appears naturally in integral geometry, namely in the algebra of unitarily invariant valuations on complex space forms. - Andreas Bernig, Feb 02 2020

			G.f. = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2530*x^6 + 16965*x^7 + ...

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.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
Handbook of Combinatorics, North-Holland '95, p. 891.
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).
W. T. Tutte, The enumerative theory of planar maps, in A Survey of Combinatorial Theory (J. N. Srivastava et al. eds.), pp. 437-448, North-Holland, Amsterdam, 1973.

T. D. Noe, Table of n, a(n) for n = 0..200
E. A. Bender and N. C. Wormald, The number of loopless planar maps, Discr. Math. 54:2 (1985), 235-237.
Andreas Bernig, Unitarily invariant valuations and Tutte's sequence, arXiv:2001.03372 [math.DG], 2020.
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See pp. 10-11.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
T. Daniel Brennan, Christian Ferko, and Savdeep Sethi, A Non-Abelian Analogue of DBI from $T \overline{T}$, arXiv:1912.12389 [hep-th], 2019. See also SciPost Phys. (2020) Vol. 8, 052.
Enrica Duchi and Corentin Henriet, A bijection between Tamari intervals and extended fighting fish, arXiv:2206.04375 [math.CO], 2022.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
Frédéric Chapoton, Sur le nombre d'intervalles dans les treillis de Tamari, arXiv:math/0602368 [math.CO], 2006.
Grégory Chatel, Vincent Pilaud, and Viviane Pons, The weak order on integer posets, arXiv:1701.07995 [math.CO], 2017.
Grégory Chatel and Viviane Pons, Counting smaller elements in the Tamari and m-Tamari lattices, arXiv preprint arXiv:1311.3922 [math.CO], 2013.
François David, A model of random surfaces with nontrivial critical behavior, Nuclear Physics B, v. 257 (1985), 543-576.
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
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)
Wenjie Fang, Planar triangulations, bridgeless planar maps and Tamari intervals, arXiv:1611.07922 [math.CO], 2016.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
C. Germain and J. Pallo, The number of coverings in four Catalan lattices, Intern. J. Computer Math., Vol. 61 (1996) pp. 19-28. (See p. 27.)
Hsien-Kuei Hwang, Mihyun Kang, and 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.
Zhaoxiang Li and Yanpei Liu, Chromatic sums of general maps on the sphere and the projective plane, Discr. Math. 307 (2007), 78-87.
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
K. A. Penson, K. Górska, A. Horzela, and G. H. E. Duchamp, Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution, arXiv:2209.06574 [math.CO], 2022.
Viviane Pons, Combinatorics of the Permutahedra, Associahedra, and Friends, arXiv:2310.12687 [math.CO], 2023. See p. 37.
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38. See Eq. 5.12.
William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), no. 2, 105-126. MR0572468 (81j:05073).
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, eq. (6).
Noam Zeilberger, A sequent calculus for the Tamari order, arXiv:1701.02917 [cs.LO], 2017.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], 2018-2019 (A revised version of a 2017 conference paper).
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. See also Part 2.

Row sums of A342981.
Column 0 of A146305 and A341856; Column 2 of A255918.
Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A000256, A006013, A027836, A069271, A263191, A290326.

[Binomial(4*n+1, n+1)-9*Binomial(4*n+1, n-1): n in [0..25]]; // Vincenzo Librandi, Nov 24 2016

A000260 := proc(n)
    2*(4*n+1)!/((n+1)!*(3*n+2)!) ;
end proc:

Table[Binomial[4n+1,n+1]-9*Binomial[4n+1,n-1],{n,0,25}] (* Harvey P. Dale, Aug 23 2011 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 3/4, 1, 5/4}, {4/3, 5/3, 2}, 256/27 x], {x, 0, n}]; (* Michael Somos, Dec 23 2014 *)
terms = 22; G[] = 0; Do[G[x] = 1+x*G[x]^4 + O[x]^terms, terms];
CoefficientList[(2-G[x])*G[x]^2, x] (* Jean-François Alcover, Jan 13 2018, after Mark van Hoeij *)

{a(n) = if( n<0, 0, 2 * (4*n + 1)! / ((n + 1)! * (3*n + 2)!))}; /* Michael Somos, Sep 07 2005 */

{a(n) = binomial( 4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2))}; /* Michael Somos, Mar 28 2012 */

def a(n):
    n = ZZ(n)
    return (4*n + 2).binomial(n + 1) // ((2*n + 1) * (3*n + 2))
# F. Chapoton, Aug 06 2015

a(n) = 2*(4*n+1)! / ((n+1)!*(3*n+2)!) = binomial(4*n+1, n+1) - 9*binomial(4*n+1, n-1).
G.f.: (2-g)*g^2 where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 10 2011
G.f.: hypergeom([1,1/2,3/4,5/4],[2,4/3,5/3],256*x/27) = 1 + 120*x/(Q(0)-120*x); Q(k) = 8*x*(2*k+1)*(4*k+3)*(4*k+5) + 3*(k+2)*(3*k+4)*(3*k+5) - 24*x*(k+2)*(2*k+3)*(3*k+4)*(3*k+5)*(4*k+7)*(4*k+9)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2011
a(n) = binomial(4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2)). - Michael Somos, Mar 28 2012
a(n) * (n+1) = A069271(n). - Michael Somos, Mar 28 2012
0 = F(a(n), a(n+1), ..., a(n+8)) for all n in Z where a(-1) = 3/4 and F() is a polynomial of degree 2 with integer coefficients and 29 monomials. - Michael Somos, Dec 23 2014
D-finite with recurrence: 3*(3*n+2)*(3*n+1)*(n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 21 2015
a(n) = Sum_{k=1..A000108(n)} k * A263191(n,k). - Alois P. Heinz, Nov 16 2015
a(n) ~ 2^(8*n+7/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n+5/2)). - Vaclav Kotesovec, Feb 26 2016
E.g.f.: 3F3(1/2,3/4,5/4; 4/3,5/3,2; 256*x/27). - Ilya Gutkovskiy, Feb 01 2017
From Gheorghe Coserea, Aug 17 2017: (Start)
G.f. y(x) satisfies:
0 = x^3*y^4 + 3*x^2*y^3 + x*(8*x+3)*y^2 - (20*x-1)*y + 16*x-1.
0 = x*(256*x - 27)*deriv(y,x) - 8*x^2*y^3 - 25*x*y^2 + 4*(24*x-11)*y + 44.
(End)
From Karol A. Penson, Apr 06 2022: (Start)
a(n) = Integral_{x=0...256/27} x^n*W(x), where
W(x) = (sqrt(2)/Pi)*(h1(x) - h2(x) + h3(x)) and
h1(x) = 3F2([-6/12,-2/12,  2/12], [ 3/12,  9/12], (27*x)/256)/((x/2)^(1/2));
h2(x) = 3F2([-3/12, 1/12,  5/12], [ 6/12, 15/12], (27*x)/256)/(x^(1/4));
h3(x) = 3F2([ 3/12, 7/12, 11/12], [18/12, 21/12], (27*x)/256)/(x^(-1/4)*32).
This integral representation is unique as the solution of n-th Hausdorff power moment of the function W. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 256/27. (End)
a(n) = (1/27^n) * Product_{1 <= i <= j <= 3*n} (3*i + j + 3)/(3*i + j - 1). Cf. A006013. - Peter Bala, Feb 21 2023

Edited by F. Chapoton, Feb 03 2011

A085357 Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Jun 25 2003

nonn

The n-th runs of ones is given by: 3 - A003849(n) (infinite Fibonacci word) = A076662(n+1). Runs of zeros are given by: A085358 and are also directly related to the Fibonacci sequence. Coefficients of A(x)^3 are found in A085359.
a(n) = 0 iff some binary digit of n is 1 while the corresponding binary digit of 3*n is 0. - Robert Israel, Jul 12 2016
The Run Length Transform of [0,1,0,0,0,...], A063524, the characteristic function of 1. (See A227349 for the definition). - Antti Karttunen, Oct 15 2016

Robert Israel, Table of n, a(n) for n = 0..10000
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen, A. Petersen and G. Skordev, Automaticity of double sequences generated by one-dimensional linear cellular automata, Theoret. Comput. Sci. 186 (1997), 195-209.
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Discrete Appl. Math. 66 (1996), 1-22.
R. Bacher and C. Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238, DOI: 10.1007/978-3-642-23283-1_15.
Index entries for characteristic functions
Index entries for sequences computed with run length transform

Cf. A001764 (ternary trees), A085358 (runs of zeros), A076662 (runs of ones), A003849 (infinite Fibonacci word), A085359 (A(x)^3).
Cf. also A003714, A005940, A008966, A063524, A227349, A277010, A324964.
Absolute values of A132971.

[Binomial(3*n,n) mod 2: n in [0..100]]; // Vincenzo Librandi, Jul 09 2016

f:= proc(n) local L,Lp;
  L:= convert(n,base,2);
  Lp:= convert(3*n,base,2);
  if has(L-Lp[1..nops(L)],1) then 0 else 1 fi
end proc:
map(f, [$0..100]); # Robert Israel, Jul 12 2016

Table[Mod[Binomial[3 n, n], 2], {n, 0, 120}] (* Michael De Vlieger, Jul 08 2016 *)

A085357(n) = !bitand(n,n<<1); \\ Antti Karttunen, Aug 22 2019

def A085357(n): return int(not n&(n<<1)) # Chai Wah Wu, Jun 25 2025

G.f.: 1 + x*A(x)^3 = A(x) (Mod 2); a(n) = A001764(n) (Mod 2).
a(n) = binomial(3n, n) (mod 2). Characteristic function of Fibbinary numbers (i.e. a(n)=1 iff n is in A003714). - Benoit Cloitre, Nov 15 2003
Recurrence: a(0) = 1, a(2n) = a(4n+1) = a(n), a(4n+3) = 0.
a(n-2) = A000256(n)(mod 2), for n>2. - John M. Campbell, Jul 08 2016
a(n) = A000621(n+1)(mod 2). - John M. Campbell, Jul 15 2016
a(n) = A000625(n)(mod 2). - John M. Campbell, Jul 15 2016
a(n) = A008966(A005940(1+n)). [Follows from the Run Length Transform interpretation, see also A277010.] - Antti Karttunen, Oct 15 2016
a(n) = abs(A132971(n)) = abs(A008683(A005940(1+n))). - Antti Karttunen, May 30 2017

A210664 Square array read by upwards antidiagonals: T(m,n) is the number of simple 3-connected triangulations of a closed region in the plane with m+3 given external edges and 3n+m internal edges, m>=0, n>=1.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 5, 6, 3, 1, 9, 20, 22, 12, 1, 14, 50, 85, 91, 52, 1, 20, 105, 254, 385, 408, 241, 1, 27, 196, 644, 1287, 1836, 1938, 1173, 1, 35, 336, 1448, 3696, 6630, 9120, 9614, 5929, 1, 44, 540, 2967, 9468, 20790, 34846, 46805, 49335, 30880

Offset: 0

N. J. A. Sloane, Mar 28 2012

A triangulation is simple if it contains no separating 3-cycle. There are n interior nodes and m+3 nodes on the boundary. - Andrew Howroyd, Feb 24 2021

			Array begins:
  1,  0,   1,    3,   12, ... (A000256)
  1,  2,   6,   22,   91, ...
  1,  5,  20,   85,  385, ...
  1,  9,  50,  254, 1287, ...
  1, 14, 105,  644, 3696, ...
  1, 20, 196, 1448, 9468, ...
  ...
From _Andrew Howroyd_, Feb 24 2021: (Start)
The array transposed for comparability with A341856 begins:
==================================================
n\m |   0    1    2     3      4      5      6
----+---------------------------------------------
  1 |   1    1    1     1      1      1      1 ...
  2 |   0    2    5     9     14     20     27 ...
  3 |   1    6   20    50    105    196    336 ...
  4 |   3   22   85   254    644   1448   2967 ...
  5 |  12   91  385  1287   3696   9468  22131 ...
  6 |  52  408 1836  6630  20790  58564 151146 ...
  7 | 241 1938 9120 34846 116641 353056 983664 ...
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1325
P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.

Rows m=0..3 are A000256, A000139, A341920, A341921.
Columns are A000012, A000096, A002415, A004305.
Antidiagonal sums give A341922.
Cf. A341856.

\\ here H is A000256 as g.f., U(n,m) is A341856 for m > 0.
H(n)={my(g=1+serreverse(x/(1+x)^4 + O(x*x^n) )); 2 - sqrt(serreverse(x*(2-g)^2*g^4)/x)}
U(n, m)={(3*(m+2)!*(m-1)!/(3*n+3*m+3)!)*sum(j=0, min(m, n-1), (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!))}
R(N, m)={my(g=2-H(N)); Vec(if(m==0, 1-g, g^(m+1)*subst(O(x*x^N) + sum(n=1, N, U(n,m)*x^n), x, x*g^2)))}
M(m, n=m)={Mat(vectorv(m+1, i, R(n,i-1)))}
M(7) \\ Andrew Howroyd, Feb 23 2021

From Andrew Howroyd, Feb 24 2021: (Start)
G.f. of row m > 0: R(x) satisfies g(x^2)^(m+1)*R(x*g(x^2)) = B(x^2) where g(x) is the g.f. of column 0 of A341856 and B(x) is the g.f. of column m of A341856.
G.f. of row m > 0: h(x)^(m+1)*B(x*h(x)^2) where 2-h(x) is the g.f. of A000256 and B(x) is the g.f. of column m of A341856.
(End)

Terms a(21) and beyond from Andrew Howroyd, Feb 23 2021

A000287 Number of rooted polyhedral graphs with n edges.

Original entry on oeis.org

1, 0, 4, 6, 24, 66, 214, 676, 2209, 7296, 24460, 82926, 284068, 981882, 3421318, 12007554, 42416488, 150718770, 538421590, 1932856590, 6969847486, 25237057110, 91729488354, 334589415276, 1224445617889, 4494622119424

Offset: 6

N. J. A. Sloane, Simon Plouffe

a(n) appears to be odd if and only if n = 2^k - 2 for some integer k >= 3. - Lewis Chen, May 05 2019

			G.f. = x^6 + 4*x^8 + 6*x^9 + 24*x^10 + 66*x^11 + 214*x^12 + 676*x^13 + ...

Handbook of Combinatorics, North-Holland '95, p. 892. (Gives different last term)
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).
W. T. Tutte, Three-connected planar maps. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 43--52. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971. MR0335323 (49 #105). - From N. J. A. Sloane, Jun 05 2012

Vincenzo Librandi, Table of n, a(n) for n = 6..1000
A. J. W. Duijvestijn and P. J. Federico, The number of polyhedral (3-connected planar) graphs, Math. Comp. 37 (1981), no. 156, 523-532.
Hsien-Kuei Hwang, Mihyun Kang, and 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.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
W. T. Tutte, A new branch of enumerative graph theory, Bull. Amer. Math. Soc., 68 (1962), 500-504.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
Liu Yanpei, On the number of rooted c-nets, J. Combin. Theory, B 36 (1984), 118-123.

Cf. A000256.

a[6] = 1; a[n_] := a[n] = ((9*(5 - 3*n)*n - 16)*a[n-1]*((n-1)!)^2 + 2*((-1)^n*(9*n*(3*n - 17) + 160)*((n-1)!)^2 + ((2*n - 2)!)))/(2*(9*n*(3*n - 11) + 88)*((n-1)!)^2); Table[ a[n], {n, 6, 31}] (* Jean-François Alcover, Oct 04 2011, after formula *)

seq(N) = {
  my(x='x+O('x^(N+5)));
  Vec(x^2 - 2*x^3/(1+x) + x*(2*x^2-10*x-1+(1-4*x)^(3/2))/(2*(x+2)^3));
};
seq(26)
\\ test: y=Ser(seq(101))*x^6; 0 == x*(x+1)^2*(x+2)*(4*x-1)*y' + 2*(x^2-11*x+1)*(x+1)^2*y + 10*x^6
\\ Gheorghe Coserea, Sep 27 2018

a(n) = b(n-1) + 2*(-1)^n, n >= 4, where b(3)=2, b(n) = (2*(2*n)!/(n!)^2 - (27*n^2+9*n-2)*b(n-1)) / (54*n^2-90*n+32). - Sean A. Irvine, Apr 14 2010
(n - 1)*a(n) = ((3/2)*n - 21/2)*a(n-1) + (8*n - 36)*a(n-2) + ((15/2)*n - 63/2)*a(n-3) + (2*n - 7)*a(n-4). - Simon Plouffe, Feb 09 2012 [Corrected by Matthew House, Sep 03 2024]
Liu Yanpei gives another recurrence. - N. J. A. Sloane, Mar 28 2012
a(n) ~ 2^(2*n+1)/(3^5*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Jul 19 2013
From Gheorghe Coserea, Apr 15 2017: (Start)
G.f.: x^2 - 2*x^3/(1+x) + x*(2*x^2-10*x-1+(1-4*x)^(3/2))/(2*(x+2)^3).
0 = x*(x+1)^2*(x+2)*(4*x-1)*y' + 2*(x^2-11*x+1)*(x+1)^2*y + 10*x^6, where y is the g.f. (End)

More terms from Sean A. Irvine, Apr 14 2010

Librandi b-file verified by N. J. A. Sloane, Mar 29 2012

cp's OEIS Frontend

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

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A085357 Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

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Magma

Maple

Mathematica

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A210664 Square array read by upwards antidiagonals: T(m,n) is the number of simple 3-connected triangulations of a closed region in the plane with m+3 given external edges and 3n+m internal edges, m>=0, n>=1.

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PARI

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A000287 Number of rooted polyhedral graphs with n edges.

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Mathematica

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A341920 Number of simple strong triangulations of a fixed pentagon with n interior nodes.

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