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A000297
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a(n) = (n+1)*(n+3)*(n+8)/6.
(Formerly M3434 N1393)
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9
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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
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OFFSET
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-1,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = -1..1000
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 Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
M. 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.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
A. Scott, T. Delaney and V. E. Hoggatt, Jr., The tribonacci sequence, Fib. Quart., 15 (1977), 193-200.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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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)
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MAPLE
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A000297:=(z-2)**2/(z-1)**4; # Simon Plouffe in his 1992 dissertation
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000292.
Sequence in context: A225254 A008186 A008264 * A078618 A304843 A062883
Adjacent sequences: A000294 A000295 A000296 * A000298 A000299 A000300
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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