A284373 Number of distinct planar connected n-polyhexes having a minimal number of vertices.

1, 1, 1, 1, 1, 3, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 6, 3, 1, 1, 1, 7, 4, 1, 4, 2, 1, 3, 2, 1, 4, 3, 1, 9, 6, 3, 1, 2, 1, 1, 10, 7, 4, 1, 5, 4, 2, 1, 4, 3, 2, 1, 6, 4, 3, 1, 12, 9, 6, 3, 1, 2, 2, 1, 1, 13, 10, 7, 4, 1, 7, 5, 4, 2, 1, 5, 4, 3, 2, 1, 7, 6, 4, 3, 1, 15, 12, 9, 6

Offset

1,6

Links

Luca Petrone, Table of n, a(n) for n = 1..386

Eric Weisstein's World of Mathematics, Polyhex.

Mathematica

polyhexeQ[{{_Integer, _Integer} .. }] := True
polyhexeQ[_] := False
rot[p_?polyhexeQ] := {-Last[#], Plus @@ #} & /@ p
ref[p_?polyhexeQ] := {-Plus @@ #, Last[#]} & /@ p
cyclic[p_] := Module[{i = p, ans = {p}},
  While[(i = rot[i]) != p, AppendTo[ans, i]]; ans]
dihedral[p_?polyhexeQ] := Flatten[{#, ref[#]} & /@ cyclic[p], 1]
canonical[p_?polyhexeQ] :=
Sort[Map[(# - {Min[First /@ p], Min[Last /@ p]}) &, p]] allPieces[p_] := Union[canonical /@ dihedral[p]]
polyhexes[1] := {{{0, 0}}}
polyhexes[n_] :=
polyhexes[n] =
  Module[{f, a, b, fig, ans = {}},
   fig = Map[(f = #; Map[({a, b} = #; {f, {a - 1, b - 1}, f, {a + 1, b - 2}, f, {a + 2, b - 1}, f, {a + 1, b + 1}, f, {a - 1, b + 2}, f, {a - 2, b + 1}}) &, f]) &, polyhexes[n - 1]];
   fig = Partition[Partition[Flatten[fig], 2], n];
   f = Union[canonical /@ Select[Union /@ fig, Length[#] == n &]];
   While[f != {},
    ans = {ans, First[f]};
    f = Complement[f, allPieces[First[f]]]];
   Partition[Partition[Flatten[ans], 2], n]]
coord[z_] := {Re[#], Im[#]} & /@ z
atoms[p_?polyhexeQ] := Module[{a, b, v, t, u = E^(Pi I/3)}, {{a, b} = #; v = a + b u; coord[{v, v + 1, v + 1 + u, v + 2 u, v + 2 u - 1, v + u - 1}]} & /@ p]
A = {};
n = 1;
While[n <= 386,
polyhexes[n];
polyhexes[n] = Part[polyhexes[n], #] & /@ Ordering[Length[Tally[Flatten[atoms[#], 2]]] &  /@ polyhexes[n],     BinCounts[#, {Min[#], Min[#] + 1}][[1]] & [Length[Tally[Flatten[atoms[#], 2]]] &  /@ polyhexes[n]]];
A = Flatten[{A, Length[#]}] & [Length[Tally[Flatten[atoms[#], 2]]] &  /@ polyhexes[n]];
Print[A[[n]]];
n++; ]
(* Luca Petrone, Mar 25 2017, based on a program by Jaime Rangel-Mondragón *)

Crossrefs

Cf. A121149.

Sequence in context: A084795 A364097 A030184 * A104610 A138684 A132442

Adjacent sequences: A284370 A284371 A284372 * A284374 A284375 A284376

Keyword

nonn

Author

Luca Petrone, Mar 25 2017

Status

approved