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A007997
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a(n) = ceiling((n-3)(n-4)/6).
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29
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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
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OFFSET
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3,5
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COMMENTS
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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 Poincare series] (or Molien series) for H^*(S_6, F_2).
a(n+5) is the 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
Also the number of 3-part integer compositions of n - 2 that are either weakly increasing or strictly decreasing. For example, the a(5) = 1 through a(13) = 15 compositions are:
(111) (112) (113) (114) (115) (116) (117) (118) (119)
(122) (123) (124) (125) (126) (127) (128)
(222) (133) (134) (135) (136) (137)
(321) (223) (224) (144) (145) (146)
(421) (233) (225) (226) (155)
(431) (234) (235) (227)
(521) (333) (244) (236)
(432) (334) (245)
(531) (532) (335)
(621) (541) (344)
(631) (542)
(721) (632)
(641)
(731)
(821)
(End)
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REFERENCES
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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.
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LINKS
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FORMULA
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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
a(n) = (40 + 3*(n - 7)*n - 4*cos(2*n*Pi/3) - 4*sqrt(3)*sin(2*n*Pi/3))/18. - Stefano Spezia, Dec 14 2021
Sum_{n>=5} 1/a(n) = 6 - 2*Pi/sqrt(3) + 2*Pi*tanh(sqrt(5/3)*Pi/2)/sqrt(15). - Amiram Eldar, Oct 01 2022
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EXAMPLE
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For m=7 (n=12), the 12 solutions are xyz = 000 610 520 511 430 421 331 322 662 653 644 554.
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MAPLE
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x^5*(1+x^3)/((1-x)*(1-x^2)*(1-x^3));
a := n -> (n*(n-7)-2*([1, 1, -1][n mod 3 +1]-7))/6;
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a007997 n = ceiling $ (fromIntegral $ (n - 3) * (n - 4)) / 6
a007997_list = 0 : 0 : 1 : zipWith (+) a007997_list [1..]
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CROSSREFS
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Apart from initial term, same as A058212.
A218004 counts these compositions of any length.
A000009 counts strictly decreasing compositions.
A000041 counts weakly increasing compositions.
A001523 counts unimodal compositions, with complement counted by A115981.
A333149 counts neither increasing nor decreasing strict compositions.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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