Riccardo Maffucci - cp's OEIS frontend

User: Riccardo Maffucci

Riccardo Maffucci has authored 2 sequences.

A355638 Number of polyhedra (3-polytopes) of graph radius 1 on n edges.

1, 0, 1, 1, 1, 1, 2, 2, 4, 5, 7, 10, 16, 27, 42, 67, 116, 187, 329, 570, 970, 1723, 3021, 5338, 9563, 16981, 30517, 54913, 98847, 179119, 324333, 589059, 1072997, 1955207, 3573129, 6538088

Offset: 6

Riccardo Maffucci, Jul 11 2022

Data was gathered with the help of Scientific IT & Application Support (SCITAS) High Performance Computing (HPC) for the EPFL community.

			For n=6 there is only the tetrahedron, n=8 the square pyramid, n=9 the triangular bipyramid,...

R. W. Maffucci, On unigraphic 3-polytopes of radius one, arXiv:2207.02040 [math.CO], 2022.

Cf. A002840.

Needs["IGraphM`"]
ra[8]:={Square pyramid}
ra[q]=opb[ra[q-1]]
opb[setg_] :=
Prepend[DeleteDuplicatesBy[
   Flatten[Table[
     EdgeAdd[g, UndirectedEdge[x[[1]], x[[2]]],
      GraphLayout -> "TutteEmbedding"], {g, setg}, {x,
      Flatten[Table[
        Complement[Subsets[i, {2}],
         Table[{i[[j]], i[[j + 1]]}, {j, Length[i] - 1}], {{i[[1]],
           i[[-1]]}}], {i, Select[IGFaces[g], Length[#] > 3 &]}],
       1]}]], CanonicalGraph],
  If[OddQ[EdgeCount[setg[[1]]]],
   WheelGraph[EdgeCount[setg[[1]]]/2 + 3/2,
    GraphLayout -> "TutteEmbedding", ImageSize -> 25], Nothing]]

A343916 a(n) is the minimal total number of faces of a polyhedron with at least one i-gonal face for every i in 3..n.

Original entry on oeis.org

4, 5, 6, 7, 8, 10, 11, 14, 16, 19, 23, 26, 31, 36, 41, 47, 54, 61, 68, 76, 85, 94, 103, 113, 124, 135, 146, 158, 171, 184, 197, 211, 226, 241, 256, 272, 289, 306, 323, 341, 360, 379, 398, 418, 439, 460, 481, 503, 526, 549, 572, 596, 621, 646, 671, 697, 724, 751, 778

Offset: 3

Riccardo Maffucci, May 04 2021

a(n) is also the minimal number of vertices of a polyhedral (planar, 3-connected) graph with at least one vertex of degree i for every i in 3..n (proven by duality).

			If we have an n-gonal face we need at least n+1 faces in total.
For n=3 we need at least 4 faces, and the tetrahedron has 4, so a(3)=4.
For n=4, e.g., the square pyramid has at least one triangle and one quadrilateral, so a(4)=5.
For n=5, there exists a polyhedron with 6 faces, comprising at least one triangle, one quadrilateral, and one pentagon: a(5)=6.

Stefano Spezia, Table of n, a(n) for n = 3..10000
R. W. Maffucci, Constructing certain families of 3-polytopal graphs, arXiv:2105.00022 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

LinearRecurrence[{3, -4, 4, -3, 1}, {4, 5, 6, 7, 8, 10, 11, 14, 16, 19, 23, 26, 31, 36, 41, 47}, 40] (* Greg Dresden, Jun 19 2021 *)

For n >= 14, a(n) = ceiling((n^2 - 11*n + 62)/4).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n > 18. - Stefano Spezia, May 05 2021
From Greg Dresden, Jun 19 2021: (Start)
For n >= 14, a(n) = (n^2 - 11*n + 62)/4 for n == 1,2 (mod 4), and a(n) = 1/2 + (n^2 - 11*n + 62)/4 for n == 0,3 (mod 4).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n >= 26. (End)

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User: Riccardo Maffucci

A355638 Number of polyhedra (3-polytopes) of graph radius 1 on n edges.

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