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A005289 Number of graphs on n nodes with 3 cliques.

%I A005289 M3440 #45 Feb 16 2025 08:32:28
%S A005289 0,0,1,4,12,31,67,132,239,407,657,1019,1523,2211,3126,4323,5859,7806,
%T A005289 10236,13239,16906,21346,26670,33010,40498,49290,59543,71438,85158,
%U A005289 100913,118913,139398,162609,188817,218295,251349,288285
%N A005289 Number of graphs on n nodes with 3 cliques.
%D A005289 R. K. Guy, personal communication.
%D A005289 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005289 R. K. Guy, <a href="/A005289/a005289.pdf">Letter to N. J. A. Sloane, Apr 1988</a>
%H A005289 R. K. Guy, <a href="https://www.jstor.org/stable/2318549">Monthly research problems, 1969-73</a>, Amer. Math. Monthly, 80 (1973), 1120-1128.
%H A005289 R. K. Guy, <a href="https://www.jstor.org/stable/2318257">Monthly research problems, 1969-75</a>, Amer. Math. Monthly, 82 (1975), 995-1004.
%H A005289 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A005289 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005289 K. B. Reid, <a href="https://doi.org/10.1112/jlms/s2-8.1.94">The number of graphs on N vertices with 3 cliques</a>, J. London Math. Soc. (2) 8 (1974), 94-98.
%H A005289 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Clique.html">Clique.</a>
%H A005289 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-4,2,2,2,-4,-1,3,-1)
%F A005289 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
%F A005289 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
%p A005289 A005289p := proc(n)
%p A005289     n*(2*n^2+3*n-6)/72 ;
%p A005289     round(%) ;
%p A005289 end proc:
%p A005289 A005289 := proc(n)
%p A005289     if type(n,'even') then
%p A005289         n*(n^2-4)*(n^2-6)/240+A005289p(n) ;
%p A005289     else
%p A005289         n*(n^2-1)*(n^2-9)/240+A005289p(n) ;
%p A005289     end if;
%p A005289 end proc:
%p A005289 seq(A005289(n),n=1..40) ; # _R. J. Mathar_, Aug 23 2015
%t A005289 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 *)
%K A005289 nonn,nice,easy
%O A005289 1,4
%A A005289 _N. J. A. Sloane_