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Number of n-node triangulations of the torus S_1 in which every node has degree >= 6.
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%I #21 Sep 02 2018 15:27:59

%S 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,

%T 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,

%U 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

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

%H Ulrich Brehm and Wolfgang Kuhnel, Equivelar maps on the torus, <a href="http://www.igt.uni-stuttgart.de/LstDiffgeo/Kuehnel/preprints/torii.pdf">Universitat Stuttgart</a>, 2006.

%H M. Jungerman and G. Ringel, <a href="https://doi.org/10.1007/BF02414187">Minimal triangulations on orientable surfaces</a>, Acta Math. 145 (1980), 121-154.

%H Thom Sulanke, <a href="http://hep.physics.indiana.edu/~tsulanke/graphs/surftri/">Generating triangulations of surfaces (surftri)</a>, (also subpages).

%H Thom Sulanke and Frank H. Lutz, <a href="https://arxiv.org/abs/math/0610022">Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds</a>, arXiv:math/0610022 [math.CO], 2006-2007.

%t 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]];

%t T3[n_] := Length[{ToRules[Reduce[n == p^2 + p q + q^2 && 0 < q < p, {p, q}, Integers]]}];

%t T6[n_] := Boole[n >= 9 && (IntegerQ[Sqrt[n]] || IntegerQ[Sqrt[n/3]])]

%t T[n_] := Piecewise[{{DivisorSigma[1, n]/6 + T2[n]/2 + 2/3 T3[n] +

%t 5/6 T6[n] - (2 - Mod[n, 2]), n > 6}}];

%t Table[T[n], {n, 7, 100}] (* _Eric W. Weisstein_ after Brehm and Kuhnel, Aug 30 2018 *)

%K nonn

%O 6,4

%A _N. J. A. Sloane_, May 12 2007

%E Terms a(18) and beyond from Thom Sulanke added by _Ed Pegg Jr_, Aug 30 2018