0,7

Verified for n=11, 12 by Shengjun Pan and R. Bruce Richter, in "The Crossing Number of K_11 is 100", submitted. Still dubious for n >= 13.

Also the sum of the dimensions of the irreducible representations of su(3) that first occur in the [n-5]th tensor power of the tautological representation. - james dolan (jdolan(AT)math.ucr.edu), Jun 02 2003

It appears that a(n)=C(floor(n/2),2)*C(floor((n-1)/2),2). [From Paul Barry (pbarry(AT)wit.ie), Oct 02 2008]

Contribution from Paul Barry (pbarry(AT)wit.ie), Oct 02 2008: (Start)

We conjecture that this sequence is given by one half of the third coefficient

of the denominator polynomial of the n-th convergent to the g.f. of n!,

in which case the next numbers are 784,1008,1296,1620, 2025, 2475,...

Essentially sum{k=0..n, (-1)^(n-k) floor(k/2)ceiling(k/2)floor((k-1)/2)ceiling((k-1)/2)/2}. (End)

One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1#K2) = c(K1) + c(K2) always hold, where c(K) denotes the crossing number of a knot K and K1#K2 is the connected sum of two (oriented) knots K1 and K2? Theorem 1.1. Let K1, . . . ,Kn be oriented knots in the 3-sphere. Then (c(K1) + . . . + c(Kn)) / 152 <= c(K1# . . . #Kn) <= c(K1) + . . . + c(Kn). [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 26 2009]

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

P. Erdos and R. K. Guy, Crossing number problems, Amer. Math. Monthly, 80 (1973), 52-58.

R. K. Guy, The crossing number of the complete graph, Bull. Malayan Math. Soc., Vol. 7, pp. 68-72, 1960.

A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280.

T. L. Saaty, The number of intersections in complete graphs, Engrg. Cybernetics 9 (1971), no. 6, 1102-1104 (1972).; translated from Izv. Akad. Nauk SSSR Tehn. Kibernet. 1971, no. 6, 151-154 (Russian). Math. Rev. 58 #21749.

C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.

J. Dolan et al., su(3) and Zarankiewicz's conjecture

T. L. Saaty, The Minimum Number Of Intersections In Complete Graphs

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

E. Weisstein, Zarankiewicz's Conjecture.html

Drago Bokal, Gasper Fijavz and David R. Wood, The Minor Crossing Number of Graphs with an Excluded Minor, math.CO/0609707.

Marc Lackenby, The crossing number of composite knots, Aug 25, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 26 2009]

It is not known if A000241 and A028723 agree. Cf. A007333, A014540, A030179.

Cf. A121021.

Sequence in context: A093446 A132920 A127645 this_sequence A028723 A057578 A015635

Adjacent sequences: A000238 A000239 A000240 this_sequence A000242 A000243 A000244

nonn,nice

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

Bokal et al. link from Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 08 2006

New arXiv citation. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 26 2009]

ArXiv URL replaced by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 23 2009