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A000933
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Genus of complete graph on n nodes.
(Formerly M0503 N0182)
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3
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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; internal format)
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OFFSET
| 1,8
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COMMENTS
| (1+x)*(1+x^3)*(1+x^5)/((1-x^2)*(1-x^4)*(1-x^6)) is the Poincare series (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) \otimes 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.
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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.
G. Ringel and J. W. T. Youngs, Solution of the Heawood map-coloring problem, Proc. Nat. Acad. Sci. USA, 60 (1968), 438-445.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Graph Genus
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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(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=1, a(6)=1, a(7)=1, a(8)=2, a(9)=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)[From Harvey P. Dale, Dec 18 2011]
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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.
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MAPLE
| A000933:=-z**4*(1-z+z**2-z**3+z**4)/(z**2+z+1)/(1+z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| CoefficientList[Series[x^5(1+x^5)/((1-x)(1-x^3)(1-x^4)), {x, 0, 70}], x] (* or *) Join[{0, 0}, LinearRecurrence[{2, -2, 3, -3, 2, -2, 1}, {0, 0, 1, 1, 1, 2, 3}, 70]] (* From Harvey P. Dale, Dec 18 2011 *)
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PROG
| (PARI) a(n)=if(n<3, 0, ceil((n-3)*(n-4)/12)) /* Michael Somos Aug 24 2005 */
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CROSSREFS
| Cf. A007997, A128425 (primes which are the genus of some complete graph).
Sequence in context: A072666 A075471 A193687 * A036409 A005423 A067319
Adjacent sequences: A000930 A000931 A000932 * A000934 A000935 A000936
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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