A005289 Number of graphs on n nodes with 3 cliques.
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
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, Letter to N. J. A. Sloane, Apr 1988
- 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.
- Index entries for linear recurrences with constant coefficients, signature (3,-1,-4,2,2,2,-4,-1,3,-1)
Programs
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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
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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 *)
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