A000944 -id:A000944 - cp's OEIS frontend

A262391 Erroneous duplicate of A000944.

0, 0, 0, 1, 2, 6, 34, 257, 2606, 440564, 6384634, 96262938, 1496225352, 23833988129, 387591510244, 6415851530241, 107854282197058

Offset: 1

Jiazhen Tan, Sep 21 2015

dead

			For n=4 the only possible shape is a tetrahedron with four triangle sides.
For n=5 the only possible shapes are the square pyramid and triangular prism.

A000109 Number of simplicial polyhedra with n vertices; simple planar graphs with n vertices and 3n-6 edges; maximal simple planar graphs with n vertices; planar triangulations with n vertices; triangulations of the sphere with n vertices; 3-connected cubic planar graphs on 2n-4 vertices.

Original entry on oeis.org

1, 1, 1, 2, 5, 14, 50, 233, 1249, 7595, 49566, 339722, 2406841, 17490241, 129664753, 977526957, 7475907149, 57896349553, 453382272049, 3585853662949, 28615703421545

Offset: 3

N. J. A. Sloane

Every planar triangulation on n >= 4 vertices is 3-connected (the connectivity either 3, 4, or 5) and its dual graph is a 3-connected cubic planar graph on 2n-4 vertices. - Manfred Scheucher, Mar 17 2023

G. Brinkmann and Brendan McKay, in preparation. [Looking at http://users.cecs.anu.edu.au/~bdm/publications.html, there are a few papers with Brinkmann that seem relevant, in particular #126 but also #97, 81, 158. Perhaps the right one is 126.]
M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
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.
B. Grünbaum, Convex Polytopes. Wiley, NY, 1967, p. 424.
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).

David Wasserman, Table of n, a(n) for n = 3..23
J. Bokowski and P. Schuchert, Equifacetted 3-spheres as topes of nonpolytopal matroid polytopes, Discrete Comput. Geom. 13 (1995), no. 3-4, 347-361.
R. Bowen and S. Fisk, Generation of triangulations of the sphere [Annotated scanned copy]
R. Bowen and S. Fisk, Generation of triangulations of the sphere, Math. Comp., 21 (1967), 250-252.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs
Aharon Davidson, From Planck Area to Graph Theory: Topologically Distinct Black Hole Microstates, arXiv:1907.03090 [gr-qc], 2019.
M. Deza, M. Dutour and P. W. Fowler, Zigzags, railroads and knots in fullerenes, J. Chem. Inf. Comput. Sci., 44 (2004), 1282-1293.
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)
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
Firsching, Moritz Realizability and inscribability for simplicial polytopes via nonlinear optimization. Math. Program. 166, No. 1-2 (A), 273-295 (2017). Table 1
Komei Fukuda, Hiroyuki Miyata, and Sonoko Moriyama, Complete Enumeration of Small Realizable Oriented Matroids. Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. Also arXiv:1204.0645 [math.CO], 2012. - From _N. J. A. Sloane_, Feb 16 2013
Jan Goedgebeur and Patric R. J. Ostergard, Switching 3-Edge-Colorings of Cubic Graphs, arXiv:2105:01363 [math.CO], May 2021. See Table 4.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Lee Zheng Han and Chia Vui Leong, The Walk of Maximal Planar Graphs, 2018.
Jorik Jooken, Computer-assisted graph theory: a survey, arXiv:2508.20825 [math.CO], 2025. See p. 5.
Paul Jungeblut, Edge Guarding Plane Graphs, Master Thesis, Karlsruhe Institute of Technology (Germany, 2019).
J. Lederberg, Dendral-64, II, Report to NASA, Dec 1965 [Annotated scanned copy]
J. Lederberg, Hamilton circuits of convex trivalent polyhedra (up to 18 vertices), Am. Math. Monthly, 74 (1967), 522-527.
J. Lederberg, Hamilton circuits of convex trivalent polyhedra (up to 18 vertices), Am. Math. Monthly, 74 (1967), 522-527. (Annotated scanned copy)
F. H. Lutz, Triangulated manifolds with few vertices: Combinatorial Manifolds, arXiv:math/0506372 [math.CO], 2005.
G. P. Michon, Counting Polyhedra
Manfred Scheucher, Hendrik Schrezenmaier, and Raphael Steiner, A Note On Universal Point Sets for Planar Graphs, arXiv:1811.06482 [math.CO], 2018.
I. Sciriha and P. W. Fowler, Nonbonding Orbitals in Fullerenes: Nuts and Cores in Singular Polyhedral Graphs, J. Chem. Inf. Model., 47, 5, 1763 - 1775, 2007.
A. Stoimenow, A theorem on graph embedding with a relation to hyperbolic volume, Combinatorica, October 2016, Volume 36, Issue 5, pp 557-589.
Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.
William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), 105-126.
Eric Weisstein's World of Mathematics, Cubic Polyhedral Graph
Eric Weisstein's World of Mathematics, Simple Polyhedron
Eric Weisstein's World of Mathematics, Triangulated Graph
Index entries for "core" sequences

Cf. A005964, A007816, A058378, A253882.

Cf. A000944, A007021, A111358.

From William P. Orrick, Apr 07 2021: (Start)
a(n) >= A007816(n-3)/n! = binomial(n,2)*(4*n-11)!/(n!*(3*n-6)!) for all n >= 4.
a(n) ~ A007816(n-3)/n! = binomial(n,2)*(4*n-11)!/(n!*(3*n-6)!) ~ (1/64)*sqrt(1/(6*Pi))*n^(-7/2)*(256/27)^(n-2), using the theorem that the automorphism group of a maximal planar graph is almost certainly trivial as n gets large. (Tutte)
(End)

Extended by Brendan McKay and Gunnar Brinkmann using their program "plantri", Dec 19 2000

Definition clarified by Manfred Scheucher, Mar 17 2023

A002840 Number of polyhedral graphs with n edges.

Original entry on oeis.org

1, 0, 1, 2, 2, 4, 12, 22, 58, 158, 448, 1342, 4199, 13384, 43708, 144810, 485704, 1645576, 5623571, 19358410, 67078828, 233800162, 819267086, 2884908430, 10204782956, 36249143676, 129267865144, 462669746182, 1661652306539, 5986979643542

Offset: 6

N. J. A. Sloane

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
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. R. S. Walsh, personal communication.

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
A. J. W. Duijvestijn and P. J. Federico, The number of polyhedral (3-connected planar) graphs, Math. Comp. 37 (1981), no. 156, 523-532.
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
G. P. Michon, Counting Polyhedra - Numericana
Hugo Pfoertner, Unlabeled 3-connected planar graphs for n<=20 edges, list in PARI-readable format.
Eric Weisstein's World of Mathematics, Polyhedral Graph
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices

Column sums of A049337.

Cf. A002841, A000944, A046091, A338511, A343869, A343871.

\\ It is assumed that the 3cp.gp file (from the linked zip archive) has been read before, i.e., \r [path]3cp.gp
for(k=6,#ThreeConnectedData,print1(#ThreeConnectedData[k],", "));
\\ printing of the edge lists of the graphs for n <= 11
print(ThreeConnectedData[6..11]) \\ Hugo Pfoertner, Feb 14 2021

a(30)-a(35) from the Numericana link added by Andrey Zabolotskiy, Jun 13 2020

A007022 Number of 4-regular polyhedra with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 11, 18, 58, 139, 451, 1326, 4461, 14554, 49957, 171159, 598102, 2098675, 7437910, 26490072, 94944685, 341867921, 1236864842, 4493270976, 16387852863, 59985464681, 220320405895, 811796327750, 3000183106119

Offset: 1

N. J. A. Sloane, Apr 28 1994

Number of simple 4-regular 4-edge-connected 3-connected planar graphs; by Steinitz's theorem, every such graph corresponds to a single planar map up to orientation-reversing isomorphism. Equivalently, number of 3-connected quadrangulations of sphere with orientation-reversing isomorphisms permitted with n faces. - Andrey Zabolotskiy, Aug 22 2017

			For n=6, the sole 6-vertex 4-regular polyhedron is the octahedron. The corresponding 6-face quadrangulation is its dual graph, i. e., the cube graph.
From _Allan Bickle_, May 13 2024: (Start)
For n=8, the unique graph is the square of an 8-cycle.
For n=9, the unique graph is the dual of the Herschel graph. (End)

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

Gunnar Brinkmann, Sam Greenberg, Catherine Greenhill, Brendan D. McKay, Robin Thomas, and Paul Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54. doi:10.1016/j.disc.2005.10.005
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs
Michael B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory Series B 66:1 (1996), 87-122.
Slavik V. Jablan, Ljiljana M. Radović, and Radmila Sazdanović, Basic polyhedra in knot theory, Kragujevac J. Math. (2005) Vol. 28, 155-164.
Jorik Jooken, Computer-assisted graph theory: a survey, arXiv:2508.20825 [math.CO], 2025. See Ref. 197 at p. 5.
T. Tarnai, F. Kovács, P. W. Fowler, and S. D. Guest, Wrapping the cube and other polyhedra, Proc. Roy. Soc. A 468(2145) (2012), 2652-2666. DOI: 10.1098/rspa.2012.0116.

Cf. A000944 (all polyhedral graphs), A113204, A078672, A078666 (total number of simple 4-regular 4-edge-connected planar maps, including not 3-connected ones).

Cf. A072552, A078666, A111361, A292515 (4-regular planar graphs with restrictions).

More terms from Hugo Pfoertner, Mar 22 2003
a(29) corrected by Brendan McKay, Jun 22 2006
Leading zeros prepended by Max Alekseyev, Sep 12 2016
Offset corrected by Andrey Zabolotskiy, Aug 22 2017

A279019 Least possible number of diagonals of simple convex polyhedron with n faces.

Original entry on oeis.org

0, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450

Offset: 4

Vladimir Letsko, Dec 03 2016

Obviously, a pyramid has no diagonals. Hence minimum of diagonals of an arbitrary convex polyhedron having n faces is equal to 0.
Minimum number of diagonals among simple convex polyhedra having n faces is obtained from a polyhedron with two triangular faces, n-4 quadrangular faces and two (n-1)-sided faces. A polyhedron having 3 triangular faces, 3 pentagonal faces and 1 hexagonal face gives another example of a simple convex polyhedron with the least possible number of diagonals for n = 7. A polyhedron having 4 triangular faces and 4 hexagonal faces gives a similar example for n = 8.
Essentially the same as A103505 and A002378. - R. J. Mathar, Dec 05 2016

Colin Barker, Table of n, a(n) for n = 4..1000
David Bremner and Victor Klee, Inner Diagonals of Convex Polytopes, Journal of Combinatorial Theory, Series A, Volume 87, Issue 1, July 1999, Pages 175-197.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000944, A002378, A279015, A279022, A007531.

Table[(n-4)(n-5),{n,4,60}] (* or *) LinearRecurrence[{3,-3,1},{0,0,2},60] (* Harvey P. Dale, Sep 23 2019 *)

concat(vector(2), Vec(2*x^6 / (1 - x)^3 + O(x^60))) \\ Colin Barker, Dec 05 2016

a(n) = n^2 - 9*n + 20 = (n-4)*(n-5).
G.f.: -2*x^6/(x-1)^3. - R. J. Mathar, Dec 05 2016
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6. - Colin Barker, Dec 05 2016
E.g.f.: exp(x)*(20 - 8*x + x^2) - x^3/3 - 3*x^2 - 12*x - 20. - Stefano Spezia, Nov 24 2019
From Amiram Eldar, Jul 09 2023: (Start)
Sum_{n>=6} 1/a(n) = 1.
Sum_{n>=6} (-1)^n/a(n) = 2*log(2) - 1. (End)

A212438 Irregular triangle read by rows: T(n,k) is the number of polyhedra with n faces and k vertices (n >= 4, k=4..2n-4).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 2, 0, 0, 2, 8, 11, 8, 5, 0, 0, 2, 11, 42, 74, 76, 38, 14, 0, 0, 0, 8, 74, 296, 633, 768, 558, 219, 50, 0, 0, 0, 5, 76, 633, 2635, 6134, 8822, 7916, 4442, 1404, 233, 0, 0, 0, 0, 38, 768, 6134, 25626, 64439, 104213, 112082, 79773, 36528, 9714, 1249

Offset: 4

N. J. A. Sloane, May 16 2012

Because of duality, T(n,k) = T(k,n). - Ivan Neretin, May 25 2016

The number of edges is n+k-2. - Andrew Howroyd, Mar 27 2021

			Triangle begins:
1
0 1 1
0 1 2  2  2
0 0 2  8 11   8    5
0 0 2 11 42  74   76   38   14
0 0 0  8 74 296  633  768  558  219   50
0 0 0  5 76 633 2635 6134 8822 7916 4442 1404 233
...

Andrew Howroyd, Table of n, a(n) for n = 4..199 (rows 4..17)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Table 9-11.
G. P. Michon, Counting Polyhedra

A049337, A058787, A212438 are all versions of the same triangle.
Row sums (the same as column sums) are A000944.
Main diagonal is A002856.
Cf. A002840 (by edges), A239893.

Terms a(53) and beyond from Andrew Howroyd, Mar 27 2021

A060296 Number of regular convex polytopes in n-dimensional space, or -1 if the number is infinite.

Original entry on oeis.org

1, 1, -1, 5, 6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Mar 24 2001

			a(2) = -1 because of the regular polygons in the plane.
a(3) = 5 because in R^3 the regular convex polytopes are the 5 Platonic solids.

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.
B. Grünbaum, Convex Polytopes. Wiley, NY, 1967, p. 424.
P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.

John Baez, Platonic Solids in All Dimensions, Nov 12 2006.
Brady Haran, Pete McPartlan, and Carlo Sequin, Perfect Shapes in Higher Dimensions, Numberphile video (2016)
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000943, A000944, A053016, A063927, A093478, A093479.

PadRight[{1, 1, -1, 5, 6}, 100, 3] (* Paolo Xausa, Jan 29 2025 *)

a(n) = 3 for all n > 4. - Christian Schroeder, Nov 16 2015

A021103 Number of two-connected (or biconnected) planar graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 44, 294, 2893, 36496, 545808, 9029737, 159563559, 2952794985, 56589742050

Offset: 0

Brendan McKay

For n < 3, conventions vary: Read & Wilson set a(2) = 0, but Gagarin et al. set a(2) = 1. - Andrey Zabolotskiy, Jun 07 2023

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998. See p. 229.

A. Gagarin, G. Labelle, P. Leroux, and T. Walsh, Structure and enumeration of two-connected graphs with prescribed three-connected components, Adv. in Appl. Math. 43 (2009), no. 1, pp. 46-74. See (116) on p. 69.

Row sums of A049336.
The labeled version is A096331.
Cf. A000944 (3-connected), A002218, A003094, A005470.

a(12)-a(14) from Gilbert Labelle (labelle.gilbert(AT)uqam.ca), Jan 20 2009
Offset 0 from Michel Marcus, Jun 05 2023
a(2) changed back to 0 by Georg Grasegger and Andrey Zabolotskiy, Jun 07 2023

A049337 Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 8, 11, 8, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 11, 42, 74, 76, 38, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 74, 296, 633, 768, 558, 219, 50

Offset: 1

Brendan McKay

			Triangle begins
  0;
  0,0;
  0,0,0,0;
  0,0,0,1,0,0,0;
  0,0,0,0,0,1,1,0,0,0;
  0,0,0,0,0,0,1,2,2,...;
  ...
From _Hugo Pfoertner_, Nov 24 2020: (Start)
Transposed table:
.
                              Nodes                        Sums
       4  5  6   7   8    9    10     11     12    13  14 |A002840
  Edges-+--+--+---+---+----+-----+------+------+-----+---+|-------
   6 | 1  .  .   .   .    .     .      .      .     .   . |      1
   7 | .  .  .   .   .    .     .      .      .     .   . |      0
   8 | .  1  .   .   .    .     .      .      .     .   . |      1
   9 | .  1  1   .   .    .     .      .      .     .   . |      2
  10 | .  .  2   .   .    .     .      .      .     .   . |      2
  11 | .  .  2   2   .    .     .      .      .     .   . |      4
  12 | .  .  2   8   2    .     .      .      .     .   . |     12
  13 | .  .  .  11  11    .     .      .      .     .   . |     22
  14 | .  .  .   8  42    8     .      .      .     .   . |     58
  15 | .  .  .   5  74   74     5      .      .     .   . |    158
  16 | .  .  .   .  76  296    76      .      .     .   . |    448
  17 | .  .  .   .  38  633   633     38      .     .   . |   1342
  18 | .  .  .   .  14  768  2635    768     14     .   . |   4199
  19 | .  .  .   .   .  538  6134   6134    558     .   . |  13384
  20 | .  .  .   .   .  219  8822  25626   8822   219   . |  43708
  21 | .  .  .   .   .   50  7916  64439  64439  7916  50 | 144810
  .. | .  .  .   .   .    .    ..     ..     ..    ..  .. |     ..
     ---+--+--+---+---+----+-----+------+-------+----+---+
  Sums 1  2  7  34 257 2606 32300 440564 6384634 .. A000944
(End)

Sean A. Irvine, Table of n, a(n) for n = 1..469; rows 1..14 flattened

Cf. A000944, A002840, A021103, A003094, A049334.
A049337, A058787, A212438 are all versions of the same triangle.
Cf. A058788.

Missing zeros inserted by Sean A. Irvine, Jul 29 2021

A000943 Number of combinatorial types of simplicial n-dimensional polytopes with n+3 nodes.

Original entry on oeis.org

1, 2, 5, 8, 18, 29, 57, 96, 183, 318, 604, 1080, 2047, 3762, 7145, 13354, 25471, 48164, 92193, 175780, 337581, 647313, 1246849, 2400828, 4636375, 8956045, 17334785, 33570800, 65108045, 126355319, 245492226, 477284164, 928772631, 1808538336

Offset: 1

N. J. A. Sloane

B. Grünbaum, Convex Polytopes. Wiley, NY, 1967, p. 424.
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).

Cf. A049337, A000944.

with(numtheory); n := 50; for d from 2 to n do C(d) := 0; for h from 1 to d+3 do if (h mod 2 = 1) and (d+3 mod h = 0) then C(d) := C(d) + phi(h) * 2^((d+3)/h); fi; od; C(d) := 2^(floor(d/2)) - floor ((d+4)/2) + C(d)/(4*(d+3)); od: A000943 := n-> eval(C(n));

a[ n_ ] := 2^Floor[ n/2 ]-Floor[ (n+4)/2 ]+(1/(4*(n+3)))*Plus@@Map[ EulerPhi[ # ]*2^((n+3)/#)&, Select[ Divisors[ n+3 ], OddQ ] ]

n=12 term corrected (typo in reference), formula (due to Perles) and more terms from Lukas Finschi (finschi(AT)ifor.math.ethz.ch), Mar 06 2001

cp's OEIS Frontend

A262391 Erroneous duplicate of A000944.

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A000109 Number of simplicial polyhedra with n vertices; simple planar graphs with n vertices and 3n-6 edges; maximal simple planar graphs with n vertices; planar triangulations with n vertices; triangulations of the sphere with n vertices; 3-connected cubic planar graphs on 2n-4 vertices.

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A002840 Number of polyhedral graphs with n edges.

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A007022 Number of 4-regular polyhedra with n nodes.

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A279019 Least possible number of diagonals of simple convex polyhedron with n faces.

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A212438 Irregular triangle read by rows: T(n,k) is the number of polyhedra with n faces and k vertices (n >= 4, k=4..2n-4).

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A060296 Number of regular convex polytopes in n-dimensional space, or -1 if the number is infinite.

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Mathematica

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A021103 Number of two-connected (or biconnected) planar graphs with n nodes.

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A049337 Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.

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