login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000297 a(n) = (n+1)*(n+3)*(n+8)/6.
(Formerly M3434 N1393)
9
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

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

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).

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

CROSSREFS

Cf. A000292.

Sequence in context: A225254 A008186 A008264 * A078618 A304843 A062883

Adjacent sequences:  A000294 A000295 A000296 * A000298 A000299 A000300

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 17 08:24 EST 2018. Contains 318192 sequences. (Running on oeis4.)