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%I #41 Jun 03 2023 09:30:38
%S 1,1,2,6,17,73,389,2274,14502,97033,672781,4792530,34911786,259106122,
%T 1954315346,14949368524,115784496932,906736988527,7171613842488,
%U 57231089062625,460428456484557,3731572377382341,30447133566946517,249968326771680542,2063931874299323140
%N Number of 3-connected planar triangulations of the sphere with n vertices up to orientation preserving isomorphisms.
%H Andrew Howroyd, <a href="/A253882/b253882.txt">Table of n, a(n) for n = 4..500</a>
%H CombOS - Combinatorial Object Server, <a href="http://combos.org/plantri">generate planar graphs</a>
%H Pascal Honvault, <a href="https://doi.org/10.11575/cdm.v16i1.62636">Equivalent classes of degree sequences for triangulated polyhedra and their convex realization</a>, Contributions to Disc. Math. (2021) Vol. 16, No. 1, 128-137.
%H Pascal Honvault, <a href="https://hal.science/hal-03744217">Local geometry of polyhedra</a>, hal-03744217 [math], 2022.
%H The House of Graphs, <a href="https://houseofgraphs.org/meta-directory/planar">Planar graphs</a>
%o (PARI) a(n)={if(n<3, 0, (2*binomial(4*(n-3)+1, n-3)/((n-2)*(3*n-7))
%o + 3*sumdiv(n-2, d, if(d>=2, my(s=(n-2)/d); eulerphi(d)*binomial(4*s,s))/4)
%o + if(n%2==1, my(s=(n-3)/2); 3*binomial(4*s,s)*(2*s+1)/(3*s+1))
%o + if(n%3==1, my(s=(n-4)/3); 8*binomial(4*s,s)*(4*s+1)/(3*s+1))
%o + if(n%3==0, my(s=(n-3)/3); 2*binomial(4*s,s)) )/(6*(n-2)))} \\ _Andrew Howroyd_, Mar 02 2021
%Y Cf. A000109 (full automorphism group), A000260 (rooted at an edge), A000944, A002709 (with a distinguished face).
%K nonn
%O 4,3
%A _Danny Rorabaugh_, Feb 27 2015
%E Name clarified and terms a(24) and beyond from _Andrew Howroyd_, Mar 02 2021
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