A175891
Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k)|0
Original entry on oeis.org
1, 1, 5, 29, 185, 1226, 8553, 61642, 455337, 3429002, 26229691, 203237747, 1591820564, 12582288455, 100241042348, 804090987555, 6488942266564, 52644171729304, 429123506792664, 3512829202462126, 28866426741057006, 238031465396515626, 1969001793889730276
Offset: 0
-
b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, add(b(x-j, y)+b(x, y-j), j=1..4)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..35); # Alois P. Heinz, May 16 2017
-
b[x_, y_] := b[x, y] = If[y > x || y < 0, 0, If[x == 0, 1, Sum[b[x - j, y] + b[x, y - j], {j, 1, 4}]]];
a[n_] := b[n, n];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)
A007028
Number of bipartite polyhedral graphs with n nodes.
Original entry on oeis.org
1, 0, 1, 1, 5, 5, 21, 40, 176, 500, 2053, 7532, 31206, 124552, 521332, 2167599, 9183879, 38965327, 166956753, 718004565, 3107450143, 13502835176, 58956369552, 258398671525
Offset: 8
- 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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- 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
- M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory, Series B, Volume 66, Issue 1, January 1996, Pages 87-122.
- Iva Kodrnja and Helena Koncul, Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N)), arXiv:2405.10747 [math.NT], 2024. See p. 10.
A119501
Number of isomorphism classes of 3-connected simple planar graphs (convex polytopes) where isomorphism does not allow reflection.
Original entry on oeis.org
0, 0, 0, 1, 2, 8, 45, 419, 4798, 62754, 872411, 12728018, 192324654, 2991463239, 47663036427, 775158142233, 12831576165782
Offset: 1
- 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
A111357
Numbers of planar triangulations with minimum degree 5 and without separating 3-cycles - that is 3-cycles where the interior and exterior contain at least one vertex.
Original entry on oeis.org
1, 0, 1, 1, 3, 4, 12, 23, 73, 191, 649, 2054, 7209, 24963, 89376, 320133, 1160752, 4218225, 15414908, 56474453, 207586410, 764855802, 2825168619, 10458049611, 38795658003, 144203518881, 537031911877, 2003618333624, 7488436558647
Offset: 12
The icosahedron is the smallest triangulation with minimum degree 5 and it doesn't contain any separating triangles. Examples can easily be seen as 2D and 3D pictures using the program CaGe cited above.
A262322
The number of 4-connected triangulations of the triangle with n inner vertices.
Original entry on oeis.org
1, 0, 0, 1, 1, 3, 13, 47, 217, 1041, 5288, 27844, 150608, 831229
Offset: 0
- 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]
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