A253882 Number of 3-connected planar triangulations of the sphere with n vertices up to orientation preserving isomorphisms.

1, 1, 2, 6, 17, 73, 389, 2274, 14502, 97033, 672781, 4792530, 34911786, 259106122, 1954315346, 14949368524, 115784496932, 906736988527, 7171613842488, 57231089062625, 460428456484557, 3731572377382341, 30447133566946517, 249968326771680542, 2063931874299323140

Offset: 4

Author

Danny Rorabaugh, Feb 27 2015

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 4..500
• CombOS - Combinatorial Object Server, generate planar graphs
• Pascal Honvault, Equivalent classes of degree sequences for triangulated polyhedra and their convex realization, Contributions to Disc. Math. (2021) Vol. 16, No. 1, 128-137.
• Pascal Honvault, Local geometry of polyhedra, hal-03744217 [math], 2022.
• The House of Graphs, Planar graphs

Crossrefs

Cf. A000109 (full automorphism group), A000260 (rooted at an edge), A000944, A002709 (with a distinguished face).

Programs

PARI

a(n)={if(n<3, 0, (2*binomial(4*(n-3)+1, n-3)/((n-2)*(3*n-7))
  + 3*sumdiv(n-2, d, if(d>=2, my(s=(n-2)/d); eulerphi(d)*binomial(4*s,s))/4)
  + if(n%2==1, my(s=(n-3)/2); 3*binomial(4*s,s)*(2*s+1)/(3*s+1))
  + if(n%3==1, my(s=(n-4)/3); 8*binomial(4*s,s)*(4*s+1)/(3*s+1))
  + if(n%3==0, my(s=(n-3)/3); 2*binomial(4*s,s)) )/(6*(n-2)))} \\ Andrew Howroyd, Mar 02 2021

Extensions

Name clarified and terms a(24) and beyond from Andrew Howroyd, Mar 02 2021