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%I #18 Sep 22 2021 03:09:52
%S 1,0,1,3,21,586,50531,16141671
%N Number of genus-surfaces with n vertices.
%H Gennaro Amendola, <a href="https://arxiv.org/abs/0705.1835">Decomposition and Enumeration of Triangulated Surfaces</a>, Experiment. Math. 17-2 (2008), 153-166; arXiv:0705.1835 [math.CO], 2007, table 2, page 20.
%e a(3) = 1 because the unique surface with 3 vertices is on the closed surface S^2.
%e a(4) = 0 because there are no surface with 4 vertices in Amendola's table.
%e a(5) = 1 because the unique surface with 5 vertices is on the closed surface RP^2.
%e a(6) = 3 because there is a unique surface with 6 vertices on the closed surface T^2 and two on RP^2, so 1 + 2 = 3.
%e a(7) = 21 because there are 5 surfaces with 7 vertices on the closed surface T^2, 6 on RP^2 and 10 on K^2, so 5 + 6 + 10 = 21.
%e a(8) = 46 + 11 + 108 + 284 + 134 + 3 = 586 (see table).
%e a(9) = 230 + 1261 + 59 + 28 + 597 + 6919 + 18166 + 18199 + 4994 + 78 = 50531 (see table).
%e a(10) = 1513 + 50878 + 99177 + 3892 + 356 + 3864 + 82588 + 713714 + 3006044 + 5672821 + 4999850 + 1453490 + 53484 = 16141671.
%Y Cf. A108239.
%K nonn,more
%O 3,4
%A _Jonathan Vos Post_, Dec 06 2007
%E Edited by _N. J. A. Sloane_, Dec 07 2007
%E Missing a(5) = 1 inserted by _Andrey Zabolotskiy_, Nov 20 2017
%E Name corrected by _Andrey Zabolotskiy_, Sep 21 2021
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