A375617 Numbers of facially complete 2-connected planar embeddings.

0, 0, 1, 3, 6, 15, 32, 94, 295, 1169, 4870, 22110, 102490, 489479, 2370856, 11655722, 57918613, 290697549, 1471349079, 7504192109, 38532719288, 199076246027, 1034236802988, 5400337234593, 28329240686563, 149244907924935, 789351357094770, 4190055030317638

Offset: 1

Author

Eric W. Weisstein, Aug 21 2024

Keywords

• nonn

Links

• Eric W. Weisstein, Table of n, a(n) for n = 1..100
• James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See pp. 7,11.
• Eric Weisstein's World of Mathematics, Facially Complete Planar Embedding.

Programs

Mathematica

prism[n_] := Floor[((n - 3)^2 + 6)/12]
tetrahedral[n_] := prism[n - 1]
bipartite[n_] := prism[n - 2]
wheel[n_] := (Mod[n - 1, 2] + 3) 2^Quotient[n - 1, 2]/4 + DivisorSum[n - 1, EulerPhi[#] 2^((n - 1)/#) &]/(2 (n - 1)) - 3
cycle[n_] := Module[{f, F, x},
  f[x_, m_] := x + Sum[(Binomial[s - 2, r - 1] Binomial[r + s - 1, s] x^s)/r, {r, m}, {s, 2, m}];
  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}];
  CoefficientList[F[x, n], x][[-1]]]
a[1] = a[2] = 0;
a[n_] := prism[n] + tetrahedral[n] + bipartite[n] + wheel[n] + cycle[n]
Table[a[n], {n, 20}]

Formula

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