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A000933 Genus of complete graph on n nodes.
(Formerly M0503 N0182)
10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
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.
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
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
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
EXAMPLE
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 + ...
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 *)
PROG
(PARI) {a(n) = if( n<3, 0, ceil((n-3) * (n-4) / 12))}; /* Michael Somos, Aug 24 2005 */
(Magma) [n le 2 select 0 else Ceiling(Binomial(n-3, 2)/6): n in [1..70]]; // G. C. Greubel, Dec 08 2022
(SageMath) [0, 0]+[ceil(binomial(n-3, 2)/6) for n in range(3, 71)] # G. C. Greubel, Dec 08 2022
CROSSREFS
Cf. A007997.
Sequence in context: A072666 A075471 A193687 * A253012 A036409 A005423
KEYWORD
easy,nonn,nice
AUTHOR
STATUS
approved

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Last modified March 19 07:49 EDT 2024. Contains 370958 sequences. (Running on oeis4.)