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URL: https://oeis.org/A005289

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A005289
Number of graphs on n nodes with 3 cliques.
1
0, 0, 1, 4, 12, 31, 67, 132, 239, 407, 657, 1019, 1523, 2211, 3126, 4323, 5859, 7806, 10236, 13239, 16906, 21346, 26670, 33010, 40498, 49290, 59543, 71438, 85158, 100913, 118913, 139398, 162609, 188817, 218295, 251349, 288285
OFFSET
1,4
REFERENCES
R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
R. K. Guy, Monthly research problems, 1969-73, Amer. Math. Monthly, 80 (1973), 1120-1128.
R. K. Guy, Monthly research problems, 1969-75, Amer. Math. Monthly, 82 (1975), 995-1004.
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
K. B. Reid, The number of graphs on N vertices with 3 cliques, J. London Math. Soc. (2) 8 (1974), 94-98.
Eric Weisstein's World of Mathematics, Clique.
FORMULA
G.f.: x^3*(1+x+3*x^3+x^2) / ( (1+x+x^2)*(1+x)^2*(x-1)^6 ). - Simon Plouffe in his 1992 dissertation
288*a(n) = -4*n^3+12*n^2-21*n/5-14+6*n^5/5+(-1)^n*9*(n-2) +32*A057078(n). - R. J. Mathar, Jul 30 2024
MAPLE
A005289p := proc(n)
n*(2*n^2+3*n-6)/72 ;
round(%) ;
end proc:
A005289 := proc(n)
if type(n, 'even') then
n*(n^2-4)*(n^2-6)/240+A005289p(n) ;
else
n*(n^2-1)*(n^2-9)/240+A005289p(n) ;
end if;
end proc:
seq(A005289(n), n=1..40) ; # R. J. Mathar, Aug 23 2015
MATHEMATICA
s = x^2*(3*x^3+x^2+x+1) / ((x-1)^6*(x+1)^2*(x^2+x+1)) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, Nov 27 2015 *)
LinearRecurrence[{3, -1, -4, 2, 2, 2, -4, -1, 3, -1}, {0, 0, 1, 4, 12, 31, 67, 132, 239, 407}, 40] (* Harvey P. Dale, Oct 18 2025 *)
CROSSREFS
Sequence in context: A297079 A074210 A299053 * A037255 A027658 A001982
KEYWORD
nonn,nice,easy
STATUS
approved