A129033 Number of n-node triangulations of the torus S_1 in which every node has degree >= 6.

0, 1, 1, 2, 1, 1, 4, 2, 2, 4, 5, 2, 5, 3, 6, 6, 4, 3, 11, 5, 5, 7, 9, 4, 11, 5, 11, 8, 7, 8, 16, 6, 8, 10, 16, 6, 15, 7, 13, 14, 10, 7, 24, 10, 14, 12, 16, 8, 19, 12, 21, 14, 13, 9, 30, 10, 14, 19, 23, 14, 23, 11, 20, 16, 23, 11, 36, 12, 17, 22, 23, 16, 27, 13, 34, 21, 19, 13, 40, 18, 20, 20, 31, 14, 39, 20, 27, 22, 22, 20, 47, 16, 27, 27, 37

Offset

6,4

Comments

From Brahadeesh Sankarnarayanan, Jan 20 2026: (Start)

Here, the 1-skeleton of the triangulation is a simple graph. For triangulations by multigraphs with loops, see A003051.

By Euler's formula, any triangulation of S_1 with minimum degree 6 is in fact 6-regular. (End)

Links

Table of n, a(n) for n=6..100.

Ulrich Brehm and Wolfgang Kuhnel, Equivelar maps on the torus, Eur. J. Comb. 29 (2008) 1843-1861.

M. Jungerman and G. Ringel, Minimal triangulations on orientable surfaces, Acta Math. 145 (1980), 121-154.

Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).

Thom Sulanke and Frank H. Lutz, Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds, Eur. J. Comb. 30 (2009) 1965--1979; see also arXiv:math/0610022 [math.CO], 2006-2007.

Formula

a(n) = A003051(n) - A131743(n-1) - 2. - Brahadeesh Sankarnarayanan, Jan 20 2026

Mathematica

T2[n_] := Piecewise[{{DivisorSigma[0, n] - 2 - T6[n], Mod[n, 2] == 1}, {DivisorSigma[0, n/2] - 2, Mod[n, 4] == 2}}, DivisorSigma[0, n/2] + DivisorSigma[0, n/4] - 4 - T6[n]];
T3[n_] := Length[{ToRules[Reduce[n == p^2 + p q + q^2 && 0 < q < p, {p, q}, Integers]]}];
T6[n_] := Boole[n >= 9 && (IntegerQ[Sqrt[n]] || IntegerQ[Sqrt[n/3]])]
T[n_] := Piecewise[{{DivisorSigma[1, n]/6 + T2[n]/2 + 2/3 T3[n] +
  5/6 T6[n] - (2 - Mod[n, 2]), n > 6}}];
Table[T[n], {n, 7, 100}] (* Eric W. Weisstein after Brehm and Kuhnel, Aug 30 2018 *)

Crossrefs

Cf. A003051, A131743.

Sequence in context: A244581 A064191 A127420 * A054090 A239456 A122517

Adjacent sequences: A129030 A129031 A129032 * A129034 A129035 A129036

Keyword

nonn

Author

N. J. A. Sloane, May 12 2007

Extensions

Terms a(18) and beyond from Thom Sulanke added by Ed Pegg Jr, Aug 30 2018

Status

approved