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A007997 a(n) = ceiling((n-3)(n-4)/6). 18
0, 0, 1, 1, 2, 4, 5, 7, 10, 12, 15, 19, 22, 26, 31, 35, 40, 46, 51, 57, 64, 70, 77, 85, 92, 100, 109, 117, 126, 136, 145, 155, 166, 176, 187, 199, 210, 222, 235, 247, 260, 274, 287, 301, 316, 330, 345, 361, 376, 392, 409, 425, 442, 460, 477, 495, 514, 532, 551, 571, 590, 610 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,5

COMMENTS

Number of solutions to x+y+z=0 (mod m) with 0<=x<=y<=z<m, where m = n-5.

Nonorientable genus of complete graph on n nodes.

Also (with different offset) Molien series for alternating group A_3.

(1+x^3 ) / ((1- x)*(1-x^2)*(1-x^3)) is the Poincaré series (or Molien series) for H^*(S_6, F_2).

a(n+5) = number of necklaces with 3 black beads and n white beads.

The g.f./x^5 is Z(C_3,x), the 3-variate cycle index polynomial for the cyclic group C_3, with substitution x[i]->1/(1-x^i), i=1,2,3. Therefore by Polya enumeration a(n+5) is the number of cyclically inequivalent 3-necklaces whose 3 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... . See A102190 for Z(C_3,x). - Wolfdieter Lang, Feb 15 2005

a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x = (y mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012

REFERENCES

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004, p. 204.

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.

J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987; see \bar{I}(n) p. 221.

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 740.

E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, 2004.

LINKS

T. D. Noe, Table of n, a(n) for n = 3..1000

C. Ahmed, P. Martin, V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.

Index entries for Molien series

Index entries for sequences related to necklaces

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1)

FORMULA

a(n) = a(n-3) + n - 2, a(0)=0, a(1)=0, a(2)=1 [Offset 0]. - Paul Barry, Jul 14 2004

G.f.: x^5*(1+x^3)/((1-x)*(1-x^2)*(1-x^3)) = x^5*(1-x+x^2)/((1-x)^2*(1-x^3)).

a(n+5) = sum_{k=0..floor(n/2)} C(n-k,L(k/3)), where L(j/p) is the Legendre symbol of j and p. - Paul Barry, Mar 16 2006

a(3)=0, a(4)=0, a(5)=1, a(6)=1, a(7)=2, a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5). - Harvey P. Dale, Jan 21 2014

a(n) = (n^2 - 7*n + 14 - 2*(-1)^(2^(n + 1 - 3*floor((n+1)/3))))/6. - Luce ETIENNE, Dec 27 2014

EXAMPLE

For m=7 (n=12), the 12 solutions are xyz = 000 610 520 511 430 421 331 322 662 653 644 554.

MAPLE

x^5*(1+x^3)/((1-x)*(1-x^2)*(1-x^3));

seq(ceil(binomial(n, 2)/3), n=0..63); # Zerinvary Lajos, Jan 12 2009

a := n -> (n*(n-7)-2*([1, 1, -1][n mod 3 +1]-7))/6;

seq(a(n), n=3..64); # Peter Luschny, Jan 13 2015

MATHEMATICA

k = 3; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

Table[Ceiling[((n-3)(n-4))/6], {n, 3, 100}] (* or *) LinearRecurrence[ {2, -1, 1, -2, 1}, {0, 0, 1, 1, 2}, 100] (* Harvey P. Dale, Jan 21 2014 *)

PROG

(Haskell)

a007997 n = ceiling $ (fromIntegral $ (n - 3) * (n - 4)) / 6

a007997_list = 0 : 0 : 1 : zipWith (+) a007997_list [1..]

-- Reinhard Zumkeller, Dec 18 2013

(PARI) a(n)=(n^2-7*n+16)\6 \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

Cf. A000031, A007998, A003050, A047996, A048259, A053618.

Apart from initial term, same as A058212.

Sequence in context: A186386 A247589 A058212 * A123120 A194462 A163267

Adjacent sequences:  A007994 A007995 A007996 * A007998 A007999 A008000

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 23 08:53 EDT 2017. Contains 289686 sequences.