A000933 Genus of complete graph on n nodes.

0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 16, 18, 20, 23, 26, 29, 32, 35, 39, 43, 46, 50, 55, 59, 63, 68, 73, 78, 83, 88, 94, 100, 105, 111, 118, 124, 130, 137, 144, 151, 158, 165, 173, 181, 188, 196, 205, 213, 221, 230, 239, 248, 257, 266, 276, 286, 295, 305

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

Comments

(1+x)*(1+x^3)*(1+x^5)/((1-x^2)*(1-x^4)*(1-x^6)) is the Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=3.

Examples

			a(1)=a(2)=a(3)=a(4)=0 because K_4 is planar. a(5)=a(6)=a(7)=1 because K_7 can be embedded on the torus of genus 1.
G.f. = x^5 + x^6 + x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 5*x^11 + 6*x^12 + 8*x^13 + ...
		

References

• A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 200
• J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987; see I(n) p. 221.
• J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 740.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• G. Ringel and J. W. T. Youngs, Solution of the Heawood map-coloring problem, Proc. Nat. Acad. Sci. USA, 60 (1968), 438-445.
• Eric Weisstein's World of Mathematics, Complete Graph
• Eric Weisstein's World of Mathematics, Graph Genus
• Index entries for linear recurrences with constant coefficients, signature (2,-2,3,-3,2,-2,1).

Crossrefs

Cf. A007997.

Programs

Magma

[n le 2 select 0 else Ceiling(Binomial(n-3,2)/6): n in [1..70]]; // G. C. Greubel, Dec 08 2022

Maple

A000933:=-z**4*(1-z+z**2-z**3+z**4)/(z**2+z+1)/(1+z**2)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

Mathematica

CoefficientList[Series[x^5(1+x^5)/((1-x)(1-x^3)(1-x^4)), {x, 0, 70}], x] (* Harvey P. Dale, Dec 18 2011 *)
Join[{0, 0}, LinearRecurrence[{2, -2, 3, -3, 2, -2, 1}, {0, 0, 1, 1, 1, 2, 3}, 70]] (* Harvey P. Dale, Dec 18 2011 *)
Join[{0, 0}, Table[Ceiling[(n - 3) (n - 4)/12], {n, 3, 20}]] (* Eric W. Weisstein, Jan 19 2018 *)

PARI

{a(n) = if( n<3, 0, ceil((n-3) * (n-4) / 12))}; /* Michael Somos, Aug 24 2005 */

SageMath

[0,0]+[ceil(binomial(n-3,2)/6) for n in range(3,71)] # G. C. Greubel, Dec 08 2022

Formula

Euler transform of length 10 sequence [ 1, 0, 1, 1, 1, 0, 0, 0, 0, -1]. - Michael Somos, Aug 24 2005

G.f.: x^5*(1+x^5)/((1-x)*(1-x^3)*(1-x^4)).

a(n) = ceiling ( (n-3)*(n-4)/12 ) if n>=3.

a(n) = 2*a(n-1) - 2*a(n-2) + 3*a(n-3) - 3*a(n-4) + 2*a(n-5) - 2*a(n-6) + a(n-7) for n >= 10. - Harvey P. Dale, Dec 18 2011

G.f.: x^5*(1-x+x^2+x^4-x^3) / ((1+x^2) * (1+x+x^2) * (1-x)^3). - R. J. Mathar, Dec 18 2014

a(n) = (49 + 3*(n - 7)*n - 9*cos(n*Pi/2) - 4*cos(2*n*Pi/3) + 9*sin(n*Pi/2) - 4*sqrt(3)*sin(2*n*Pi/3))/36 for n > 2. - Stefano Spezia, Dec 14 2021