|
| |
|
|
A006918
|
|
C(n+3,3)/4, n odd; n(n+2)(n+4)/24, n even.
(Formerly M1349)
|
|
50
|
|
|
|
0, 1, 2, 5, 8, 14, 20, 30, 40, 55, 70, 91, 112, 140, 168, 204, 240, 285, 330, 385, 440, 506, 572, 650, 728, 819, 910, 1015, 1120, 1240, 1360, 1496, 1632, 1785, 1938, 2109, 2280, 2470, 2660, 2870, 3080, 3311, 3542, 3795, 4048, 4324, 4600, 4900, 5200, 5525, 5850, 6201, 6552, 6930
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Maximal number of inconsistent triples in a tournament on n nodes [Kac]
a(n-4)=number of aperiodic necklaces (Lyndon words) with 4 black beads and n-4 white beads.
Comment from Erich Friedman: also the maximum number of squares that can be formed from n lines.
Number of trees with diameter 4 where at most 2 vertices 1 away from the graph center have degree > 2. - Jon Perry, Jul 11 2003
a(n+1) is the number of partitions of n into parts of two kinds, with at most two parts of each kind. Also a(n-3) is the number of partitions of n with Durfee square of size 2. - Franklin T. Adams-Watters, Jan 27 2006
Factoring the g.f. as x/(1-x)^2 times 1/(1-x^2)^2 we find that the sequence equals (1, 2, 3, 4,...) convolved with (1, 0, 2, 0, 3, 0, 4,...), A000027 convolved with its aerated variant. - Gary W. Adamson, May 01 2009
Starting with "1" = triangle A171238 * [1,2,3,...] [From Gary W. Adamson, Dec 05 2009]
The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of A006918, e.g. Kn22(n) = a(n+1) + a(n) + 2*a(n-1) + a(n-2) and Fi2(n) = a(n) + 4*a(n-1) + a(n-2). [From Johannes W. Meijer, May 20 2011]
For n>3, a(n-4) is the number of (w,x,y,z) having all terms in {1,...,n} and w+x+y+z=|x-y|+|y-z|. [Clark Kimberling, May 23 2012]
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w+x+y < |w-x|+|x-y|. [Clark Kimberling, Jun 13 2012]
|
|
|
REFERENCES
|
J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.
S. J. Cyvin et al., Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
Eke, B. G., Monotonic triads. Discrete Math. 9 (1974), 359--363. MR0354390 (50 #6869)
John Golden and Marcus Spradlin, Collinear and Soft Limits of Multi-Loop Integrands in N= 4 Yang-Mills, Arxiv preprint arXiv:1203.1915, 2012. - From N. J. A. Sloane, Sep 14 2012
M. Kac, An example of "counting without counting", Philips Res. Reports, 30 (1975), 20*-22* [Special issue in honour of C. J. Bouwkamp]
E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, 2004.
K. B. Reid and L. W. Beineke "Tournaments", pp. 169-204 in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, p. 186 Theorem 6.11.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=0..1000
D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory
Brian OSullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=2]
Index entries for sequences related to linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).
Index entries for sequences related to Lyndon words
|
|
|
FORMULA
|
G.f.: x/((1-x)^2*(1-x^2)^2) = x/((1+x)^2*(1-x)^4).
0, 0, 0, 1, 2, 5, 8, 14, ... has a(n)=sum{k=0..n, floor(k(n-k)/2) }/2 - Paul Barry, Sep 14 2003
0, 0, 0, 0, 0, 1, 2, 5, 8, 14, 20, 30, 40, 55, ... has a(n) = binomial(floor(1/2 n), 3) + binomial(floor(1/2 n + 1/2), 3) [Eke] - N. J. A. Sloane, May 12 2012
a(0)=0, a(1)=1 a(n)=(2/(n-1))*a(n-1)+((n+3)/(n-1))*a(n-2) - Benoit Cloitre, Jun 28 2004
a(n)=floor(C(n+4, 4)/(n+4))-floor((n+2)/8)(1+(-1)^n)/2 - Paul Barry, Jan 01 2005
a(n+1) = a(n) + C([n/2]+2,2), ie, first differences are A008805. Convolution of A008619 with itself, then shifted right (or A004526 with itself, shifted left by 3). - Franklin T. Adams-Watters, Jan 27 2006
a(n+1)= (A027656(n) + A003451(n+5))/2 with a(1)=0 - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Sep 12 2008
Linear recurrence: a(n)=2a(n-1)+a(n-2)-4a(n-3)+a(n-4)+2a(n-5)-a(n-6) [From Jaume Oliver Lafont, Dec 05 2008]
Euler transform of length 2 sequence [ 2, 2]. - Michael Somos, Aug 15 2009
a(-4 - n) = -a(n).
a(n+1) + a(n) = A002623(n) [From Johannes W. Meijer, May 20 2011]
a(n) = (n+2)*(2*n*(n+4)-3*(-1)^n+3)/48 - Bruno Berselli, May 21 2011
|
|
|
EXAMPLE
|
x + 2*x^2 + 5*x^3 + 8*x^4 + 14*x^5 + 20*x^6 + 30*x^7 + 40*x^8 + 55*x^9 + ...
|
|
|
MAPLE
|
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=3)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m), m=11..58) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 09 2007
|
|
|
MATHEMATICA
|
f[n_]:=If[EvenQ[n], (n(n+2)(n+4))/24, Binomial[n+3, 3]/4]; Join[{0}, Array[f, 60]] (* From Harvey P. Dale, Apr 20 2011 *)
|
|
|
PROG
|
(PARI) { parttrees(n)=local(pt, k, nk); if (n%2==0, pt=(n/2+1)^2, pt=ceil(n/2)*(ceil(n/2)+1)); pt+=floor(n/2); for (x=1, floor(n/2), pt+=floor(x/2)+floor((n-x)/2)); if (n%2==0 && n>2, pt-=floor(n/4)); k=1; while (3*k<=n, for (x=k, floor((n-k)/2), pt+=floor(k/2); if (x!=k, pt+=floor(x/2)); if ((n-x-k)!=k && (n-x-k)!=x, pt+=floor((n-x-k)/2))); k++); pt }
(PARI) {a(n) = n += 2; (n^3 - n * (2-n%2)^2) / 24} /* Michael Somos, Aug 15 2009 */
(Haskell)
a006918 n = a006918_list !! n
a006918_list = scanl (+) 0 a008805_list
-- Reinhard Zumkeller, Feb 01 2013
|
|
|
CROSSREFS
|
Cf. A000031, A001037, A028723, A051168. a(n) = T(n, 4), array T as in A051168.
Cf. A000094.
Cf. A171238 [From Gary W. Adamson, Dec 05 2009]
Row sums of A173997 [From Gary W. Adamson, Mar 05 2010]
Sequence in context: A095348 A215725 A022907 * A165189 A011842 A000094
Adjacent sequences: A006915 A006916 A006917 * A006919 A006920 A006921
|
|
|
KEYWORD
|
nonn,nice,easy,changed
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
