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%I M3434 N1393 #100 Sep 04 2022 13:20:46
%S 0,4,12,25,44,70,104,147,200,264,340,429,532,650,784,935,1104,1292,
%T 1500,1729,1980,2254,2552,2875,3224,3600,4004,4437,4900,5394,5920,
%U 6479,7072,7700,8364,9065,9804,10582,11400,12259,13160,14104,15092,16125,17204
%N a(n) = (n+1)*(n+3)*(n+8)/6.
%C If Y and Z are 2-blocks of an n-set X then, for n>=4, a(n-5) is the number of (n-3)-subsets of X intersecting both Y and Z. - _Milan Janjic_, Nov 09 2007
%C a(n) is the number of triangles in the Turan graph T(n, n-2) for n>3. - _Robert H Cowen_, Feb 25 2018
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000297/b000297.txt">Table of n, a(n) for n = -1..1000</a>
%H Robert Cowen, <a href="https://doi.org/10.3888/tmj.20-6">Improving the Kruskal-Katona Bounds for Complete Subgraphs of a Graph</a>, The Mathematica Journal (2018) Vol. 20.
%H P. Erdős, R. K. Guy and J. W. Moon, <a href="https://doi.org/10.1112/jlms/s2-9.4.565">On refining partitions</a>, J. London Math. Soc., 9 (1975), 565-570.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A. Scott, T. Delaney and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/15-3/scott.pdf">The tribonacci sequence</a>, Fib. Quart., 15 (1977), 193-200.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: (2-x)^2 / (1-x)^4.
%F a(n) = Sum_{k=3..n} n*(k+1)/3, n>=2. - _Zerinvary Lajos_, Jan 29 2008
%F G.f.: 2*x*W(0), where W(k) = 1 + 1/( 1 - x*(k+2)*(k+4)*(k+9)/(x*(k+2)*(k+4)*(k+9) + (k+1)*(k+3)*(k+8)/W(k+1) )) ); (continued fraction). - _Sergei N. Gladkovskii_, Aug 24 2013
%F With offset 3, for n>3, a(n) = 4 binomial(n-2,2) + binomial(n-3,3), comprising the fourth column of A267633. - _Tom Copeland_, Jan 25 2016
%F From _Bob Selcoe_, Apr 02 2016 (Start):
%F a(n) = A000292(n+3) - 2n - 6.
%F a(n) = a(n-1) + (n^2 + 7n + 8)/2.
%F (End)
%p A000297:=(z-2)**2/(z-1)**4; # _Simon Plouffe_ in his 1992 dissertation
%t Table[(n + 1)*(n + 3)*(n + 8)/6, {n, -1, 100}]
%t CoefficientList[Series[x (2 - x)^2 / (1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 31 2018 *)
%o (PARI) a(n) = (n+1)*(n+3)*(n+8)/6; \\ _Altug Alkan_, Jan 10 2015
%o (GAP) List([-1..45], n->(n+1)*(n+3)*(n+8)/6); # _Muniru A Asiru_, Mar 11 2018
%o (Magma) [(n+1)*(n+3)*(n+8)/6: n in [-1..50]]; // _Vincenzo Librandi_, Oct 31 2018
%o (Python)
%o def A000297_gen(): # generator of terms
%o a, b, c = 0, 4, 4
%o while True:
%o yield a
%o a, b, c = a+b, b+c, c+1
%o it = A000297_gen()
%o A000297_list = [next(it) for _ in range(50)] # _Cole Dykstra_, Aug 05 2022
%Y Cf. A000292.
%K nonn,easy
%O -1,2
%A _N. J. A. Sloane_
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