0,3

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 (erich.friedman(AT)stetson.edu): 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 (perry(AT)globalnet.co.uk), 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. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 27 2006

Starting with offset 1 = partial sums of (1, 1, 3, 3, 6, 6, 10, 10,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 30 2009]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2009: (Start)

Equals (1, 2, 3, 4,...) convolved with (1, 0, 2, 0, 3, 0, 4,...).

a(6) = 20 = (1, 2, 3, 4, 5, 6) dot (0, 3, 0, 2, 0, 1) = (0 + 6 + 0 + 8 + 0 + 6). (End)

(1 + 2x + 5x^2 + ...) = convolution square of (1 + x + 2x^2 + 2x^3 + 3x^4 + 3x^5 + ...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. J. Cyvin et al., Polygonal systems including the corannulene ... homologs ..., Z. Naturforsch., 52a (1997), 867-873.

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.

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.

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]

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for sequences related to linear recurrences with constant coefficients

D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory

Index entries for sequences related to Lyndon words

G.f.: x/((1-x)^2*(1-x^2)^2).

0, 0, 0, 1, 2, ... has a(n)=sum{k=0..n, floor(k(n-k)/2) }/2 - Paul Barry (pbarry(AT)wit.ie), Sep 14 2003

a(0)=0, a(1)=1 a(n)=(2/(n-1))*a(n-1)+((n+3)/(n-1))*a(n-2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 28 2004

a(n)=floor(C(n+4, 4)/(n+4))-floor((n+2)/8)(1+(-1)^n)/2 - Paul Barry (pbarry(AT)wit.ie), Jan 01 2005

a(n+1) = a(n) + C([n/2]+2,2) Convolution of A008619 with itself, then shifted right (or A004526 with itself, shifted left by 3). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 27 2006

Contribution from Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Sep 12 2008: (Start)

A006918 (n+1)= (A027656 (n) + A003451(n))/2 with a(1)=0

(End)

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 (joliverlafont(AT)gmail.com), Dec 05 2008]

Euler transform of length 2 sequence [ 2, 2]. - Michael Somos Aug 15 2009

a(-4 - n) = -a(n).

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 + ...

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

(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 */

Cf. A000031, A001037, A028723, A051168. a(n) = T(n, 4), array T as in A051168.

Cf. A000094.

Adjacent sequences: A006915 A006916 A006917 this_sequence A006919 A006920 A006921

Sequence in context: A111711 A095348 A022907 this_sequence A165189 A011842 A000094

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).