0,3

Maximum value of Voronoi's principal quadratic form of the first type when variables restricted to {-1,0,1}. - Michael Somos Mar 10 2004

Comment from Emilio Apricena (emilioapricena(AT)yahoo.it), Feb 08 2009: This is the main row of a version of the "Ulam spiral" when read alternatively from left to right (see link). See also A001107, A007742, A033954, A033991. It is easy to see that the only prime in the sequence is 5.

Contribution from Mitch Phillipson, Manda Riehl, Tristan Williams (riehlar(AT)uwec.edu), Mar 06 2009: (Start)

a(n) gives the number of elements of S_2 \wr C_k that avoid the pattern 12, using the following ordering:

In S_j, a permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a<c<b. We extend this notion to S_j \wr C_n as follows. Element \psi =[ \alpha_1^\beta_1, \dots , \alpha_j^\beta_j ] avoids \tau = [ a_1 , \dots , a_m ] (\tau \in S_m) if \psi' = [ \alpha_1*\beta_1, \dots , \alpha_j*\beta_j ] avoids \tau in the usual sense. For n=2, there are 5 elements of S_2 \wr C_2 that avoid the pattern 12. They are: [ 2^1,1^1 ], [ 2^2,1^1 ], [ 2^2,1^2 ], [ 2^1,1^2 ], [ 1^2,2^1 ]

For example, if \psi = [2^1,1^2], then \psi'=[2,2] which avoids tau=[1,2] because no subsequence ab of \psi' has a<b. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 115.

Emilio Apricena, A version of the Ulam spiral

G.f.: x(1+3x)/((1+x)(1-x)^3). a(n) = n^2+n-1-[(n-1)/2].

Row sums of triangle A133983 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 30 2007

(PARI) a(n)=n^2+n-1-(n-1)\2

A011848(2n+1)=a(n).

Cf. A133983.

Sequence in context: A067253 A048822 A092390 this_sequence A091386 A164004 A025010

Adjacent sequences: A035605 A035606 A035607 this_sequence A035609 A035610 A035611

nonn

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