Search: id:A035608
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%I A035608
%S A035608 0,1,5,10,18,27,39,52,68,85,105,126,150,175,203,232,264,297,333,370,410,
%T A035608 451,495,540,588,637,689,742,798,855,915,976,1040,1105,1173,1242,1314,
%U A035608 1387,1463,1540,1620,1701,1785,1870,1958,2047,2139,2232,2328,2425
%N A035608 Expansion of x(1+3x)/((1+x)(1-x)^3).
%C A035608 Maximum value of Voronoi's principal quadratic form of the first type 
               when variables restricted to {-1,0,1}. - Michael Somos Mar 10 2004
%C A035608 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.
%C A035608 Contribution from Mitch Phillipson, Manda Riehl, Tristan Williams (riehlar(AT)uwec.edu), 
               Mar 06 2009: (Start)
%C A035608 a(n) gives the number of elements of S_2 \wr C_k that avoid the pattern 
               12, using the following ordering:
%C A035608 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 ]
%C A035608 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)
%D A035608 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 115.
%H A035608 Emilio Apricena, <a href="a035608.png">A version of the Ulam spiral</
               a>
%F A035608 G.f.: x(1+3x)/((1+x)(1-x)^3). a(n) = n^2+n-1-[(n-1)/2].
%F A035608 Row sums of triangle A133983 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 30 2007
%o A035608 (PARI) a(n)=n^2+n-1-(n-1)\2
%Y A035608 A011848(2n+1)=a(n).
%Y A035608 Cf. A133983.
%Y A035608 Sequence in context: A067253 A048822 A092390 this_sequence A091386 A164004 
               A025010
%Y A035608 Adjacent sequences: A035605 A035606 A035607 this_sequence A035609 A035610 
               A035611
%K A035608 nonn
%O A035608 0,3
%A A035608 N. J. A. Sloane (njas(AT)research.att.com).

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