|
Search: id:A006694
|
|
|
| A006694 |
|
Number of cyclotomic cosets of 2 mod 2n+1.
(Formerly M0192)
|
|
+0
28
|
|
| 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, 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, 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
(list; graph; listen)
|
|
|
OFFSET
|
0,4
|
|
|
COMMENT
|
a(0) = 0 by convention.
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.
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
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977, pp. 104-105.
|
|
LINKS
|
Ray Chandler, Table of n, a(n) for n=0..10000
|
|
FORMULA
|
Conjecture: a((3^n-1)/2)=n - Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 26 2008
|
|
EXAMPLE
|
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.
|
|
MAPLE
|
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..)];
|
|
MATHEMATICA
|
Needs["Combinatorica`"]; f[n_] := Length[ToCycles[Mod[2Range[2n], 2n + 1]]]; Table[f[n], {n, 0, 100}] (*Chandler*)
f[n_] := Length[FactorList[x^(2n + 1) - 1, Modulus -> 2]] - 2; Table[f[n], {n, 0, 100}] (*Chandler*)
|
|
CROSSREFS
|
Cf. A002326 (order of 2 mod 2n+1), A139767.
a(n) = A081844(n) - 1.
a(n) = A064286(n) + 2*A064287(n).
A001917 gives cycle counts of such permutations constructed only for odd primes.
Sequence in context: A100996 A090048 A064285 this_sequence A116595 A128315 A123566
Adjacent sequences: A006691 A006692 A006693 this_sequence A006695 A006696 A006697
|
|
KEYWORD
|
nonn,nice,easy
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com), Sep 25 2001
|
|
EXTENSIONS
|
Additional comments from Antti Karttunen Jan 05 2000
Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Apr 25 2008
Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 27 2008 at the suggestion of Ray Chandler.
|
|
|
Search completed in 0.002 seconds
|
