A006469 Number of rooted toroidal maps with 2 faces, n vertices and no isthmuses.

10, 79, 340, 1071, 2772, 6258, 12768, 24090, 42702, 71929, 116116, 180817, 273000, 401268, 576096, 810084, 1118226, 1518195, 2030644, 2679523, 3492412, 4500870, 5740800, 7252830, 9082710, 11281725, 13907124, 17022565, 20698576, 25013032, 30051648, 35908488

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A map on a torus has genus 1.

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Column 2 of A343092.

Programs

Mathematica

A006469[n_] := n*(n + 1)*(n + 2)*(n + 3)*(n*(8*n + 63) + 79)/360;
Array[A006469, 50] (* Paolo Xausa, Aug 20 2025 *)

PARI

Vec(x*(10 + 9*x - 3*x^2) / (1 - x)^7 + O(x^40)) \\ Colin Barker, Apr 22 2017

Formula

G.f.: x/(x-1)^7*(3*x^2-9*x-10). - Simon Plouffe, Master's thesis, Uqam 1992

From Colin Barker, Apr 22 2017: (Start)

a(n) = (n*(474 + 1247*n + 1215*n^2 + 545*n^3 + 111*n^4 + 8*n^5)) / 360.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.

(End)

Extensions

Name improved by Sean A. Irvine, Apr 21 2017