Search: id:A006694
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%I A006694 M0192
%S A006694 0,1,1,2,2,1,1,4,2,1,5,2,2,3,1,6,4,5,1,4,2,3,7,2,4,7,1,4,4,1,1,12,6,1,
               5,2,
%T A006694 8,7,5,2,4,1,11,4,8,9,13,4,2,7,1,2,14,1,3,4,4,5,11,8,2,7,3,18,10,1,9,10,
%U A006694 2,1,5,4,6,9,1,10,12,13,3,4,8,1,13,2,2,11,1,8,4,1,1,4,6,7,19,2,2,19,1,
               2
%N A006694 Number of cyclotomic cosets of 2 mod 2n+1.
%C A006694 a(0) = 0 by convention.
%C A006694 The number of cycles in permutations constructed from siteswap juggling 
               patterns 1, 123, 12345, 1234567, etc., i.e. the number of ball orbits 
               in such patterns minus one.
%C A006694 Conjecture: a(n) is the number of orbits in (Z\(2n+1)Z)* generated by 
               2. - Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 26 2008
%D A006694 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006694 J.-P. Allouche, Suites infinies a repetitions bornees, S\'{e}minaire 
               de Th\'{e}orie des Nombres de Bordeaux, 20 (13 April, 1984), 1-11.
%D A006694 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting 
               Codes, Elsevier/North Holland, 1977, pp. 104-105.
%H A006694 Ray Chandler, <a href="b006694.txt">Table of n, a(n) for n=0..10000</
               a>
%F A006694 Conjecture: a((3^n-1)/2)=n - Vladimir Shevelev (shevelev(AT)bgu.ac.il), 
               May 26 2008
%e A006694 Mod 15 there are 4 cosets: {1, 2, 4, 8}, {3, 6, 12, 9}, {5, 10}, {7, 
               14, 13, 11}, so a(7) = 4. Mod 13 there is only one coset: {1, 2, 
               4, 8, 3, 6, 12, 11, 9, 5, 10, 7}, so a(6) = 1.
%p A006694 with(group); with(numtheory); gen_rss_perm := proc(n) local a, i; a := 
               []; for i from 1 to n do a := [op(a), ((2*i) mod (n+1))]; od; RETURN(a); 
               end; count_of_disjcyc_seq := [seq(nops(convert(gen_rss_perm(2*j),
               'disjcyc')),j=0..)];
%t A006694 Needs["Combinatorica`"]; f[n_] := Length[ToCycles[Mod[2Range[2n], 2n 
               + 1]]]; Table[f[n], {n, 0, 100}] (*Chandler*)
%t A006694 f[n_] := Length[FactorList[x^(2n + 1) - 1, Modulus -> 2]] - 2; Table[f[n], 
               {n, 0, 100}] (*Chandler*)
%Y A006694 Cf. A002326 (order of 2 mod 2n+1), A139767.
%Y A006694 a(n) = A081844(n) - 1.
%Y A006694 a(n) = A064286(n) + 2*A064287(n).
%Y A006694 A001917 gives cycle counts of such permutations constructed only for 
               odd primes.
%Y A006694 Sequence in context: A100996 A090048 A064285 this_sequence A116595 A128315 
               A123566
%Y A006694 Adjacent sequences: A006691 A006692 A006693 this_sequence A006695 A006696 
               A006697
%K A006694 nonn,nice,easy
%O A006694 0,4
%A A006694 N. J. A. Sloane (njas(AT)research.att.com), Sep 25 2001
%E A006694 Additional comments from Antti Karttunen Jan 05 2000
%E A006694 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Apr 25 2008
%E A006694 Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 27 2008 at 
               the suggestion of Ray Chandler.

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