%I #19 Sep 21 2024 08:45:41

%S 0,0,1,3,6,15,32,94,295,1169,4870,22110,102490,489479,2370856,

%T 11655722,57918613,290697549,1471349079,7504192109,38532719288,

%U 199076246027,1034236802988,5400337234593,28329240686563,149244907924935,789351357094770,4190055030317638

%N Numbers of facially complete 2-connected planar embeddings.

%H Eric W. Weisstein, <a href="/A375617/b375617.txt">Table of n, a(n) for n = 1..100</a>

%H James Tilley, Stan Wagon, and Eric Weisstein, <a href="https://arxiv.org/abs/2409.11249">A Catalog of Facially Complete Graphs</a>, arXiv:2409.11249 [math.CO], 2024. See pp. 7,11.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FaciallyCompletePlanarEmbedding.html">Facially Complete Planar Embedding</a>.

%F a(n) = A001399(n - 6) + A001399(n - 7) + A001399(n - 8) + (A056342(n - 1) - 1) + A001004(n).

%t prism[n_] := Floor[((n - 3)^2 + 6)/12]

%t tetrahedral[n_] := prism[n - 1]

%t bipartite[n_] := prism[n - 2]

%t wheel[n_] := (Mod[n - 1, 2] + 3) 2^Quotient[n - 1, 2]/4 + DivisorSum[n - 1, EulerPhi[#] 2^((n - 1)/#) &]/(2 (n - 1)) - 3

%t cycle[n_] := Module[{f, F, x},

%t f[x_, m_] := x + Sum[(Binomial[s - 2, r - 1] Binomial[r + s - 1, s] x^s)/r, {r, m}, {s, 2, m}];

%t F[x_, m_] := Series[((3 x^2 - 2 x f[x, m] + f[x, m]^2) - (2 + 2 x + 7 x^2 - 4 x f[x, m] + 2 f[x, m]^2) f[x^2, m] + 2 f[x^2, m]^2)/(4 (2 f[x^2, m] - 1)) + Sum[If[Mod[k, d] == 0, (EulerPhi[d] f[x^d, m]^(k/d))/k, 0], {k, 3, m}, {d, k}]/2, {x, 0, m}];

%t CoefficientList[F[x, n], x][[-1]]]

%t a[1] = a[2] = 0;

%t a[n_] := prism[n] + tetrahedral[n] + bipartite[n] + wheel[n] + cycle[n]

%t Table[a[n], {n, 20}]

%K nonn

%O 1,4

%A _Eric W. Weisstein_, Aug 21 2024