A000297 a(n) = (n+1)*(n+3)*(n+8)/6. (Formerly M3434 N1393)

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

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

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.

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.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

Adjacent sequences: A000294 A000295 A000296 * A000298 A000299 A000300

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved