OFFSET
1,4
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.
FORMULA
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}]
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Eric W. Weisstein, Aug 21 2024
STATUS
approved