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A000297 a(n) = (n+1)*(n+3)*(n+8)/6.
(Formerly M3434 N1393)
10
0, 4, 12, 25, 44, 70, 104, 147, 200, 264, 340, 429, 532, 650, 784, 935, 1104, 1292, 1500, 1729, 1980, 2254, 2552, 2875, 3224, 3600, 4004, 4437, 4900, 5394, 5920, 6479, 7072, 7700, 8364, 9065, 9804, 10582, 11400, 12259, 13160, 14104, 15092, 16125, 17204 (list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
COMMENTS
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
a(n) is the number of triangles in the Turan graph T(n, n-2) for n>3. - Robert H Cowen, Feb 25 2018
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
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).
LINKS
Robert Cowen, Improving the Kruskal-Katona Bounds for Complete Subgraphs of a Graph, The Mathematica Journal (2018) Vol. 20.
P. Erdős, R. K. Guy and J. W. Moon, On refining partitions, J. London Math. Soc., 9 (1975), 565-570.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - N. J. A. Sloane, Feb 13 2013
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
A. Scott, T. Delaney and V. E. Hoggatt, Jr., The tribonacci sequence, Fib. Quart., 15 (1977), 193-200.
FORMULA
G.f.: (2-x)^2 / (1-x)^4.
a(n) = Sum_{k=3..n} n*(k+1)/3, n>=2. - Zerinvary Lajos, Jan 29 2008
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
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
From Bob Selcoe, Apr 02 2016 (Start):
a(n) = A000292(n+3) - 2n - 6.
a(n) = a(n-1) + (n^2 + 7n + 8)/2.
(End)
MAPLE
A000297:=(z-2)**2/(z-1)**4; # Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(n + 1)*(n + 3)*(n + 8)/6, {n, -1, 100}]
CoefficientList[Series[x (2 - x)^2 / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Oct 31 2018 *)
PROG
(PARI) a(n) = (n+1)*(n+3)*(n+8)/6; \\ Altug Alkan, Jan 10 2015
(GAP) List([-1..45], n->(n+1)*(n+3)*(n+8)/6); # Muniru A Asiru, Mar 11 2018
(Magma) [(n+1)*(n+3)*(n+8)/6: n in [-1..50]]; // Vincenzo Librandi, Oct 31 2018
(Python)
def A000297_gen(): # generator of terms
a, b, c = 0, 4, 4
while True:
yield a
a, b, c = a+b, b+c, c+1
it = A000297_gen()
A000297_list = [next(it) for _ in range(50)] # Cole Dykstra, Aug 05 2022
CROSSREFS
Cf. A000292.
Sequence in context: A225254 A008186 A008264 * A078618 A246988 A304843
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified April 16 23:37 EDT 2024. Contains 371756 sequences. (Running on oeis4.)