This site is supported by donations to The OEIS Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A130293 Number of necklaces of n beads with up to n colors, with cyclic permutation {1,..,n} of the colors taken to be equivalent. 15
 1, 2, 5, 20, 129, 1316, 16813, 262284, 4783029, 100002024, 2357947701, 61917406672, 1792160394049, 56693913450992, 1946195068379933, 72057594071484456, 2862423051509815809, 121439531097819321972 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS From Olivier Gérard, Aug 01 2016: (Start) Equivalent to the definition: number of classes of endofunctions of [n] under rotation and translation mod n. . Classes can be of size between n and n^2 depending on divisibility properties of n. . n   n   2n   3n  ...  n^2 -------------------------- 1   1 2   2 3   3                   2 4   4    2             14 5   5    0            124 6   6    6   22      1282 7   7    0          16806 . For prime n, the only possible class sizes are n and n^2, the classes of size n are the n arithmetical progression modulo n so #(c-n)=n, #(c-n^2)=(n^n - n*n)/n^2 = n^(n-2)-1 and a(n)= n^(n-2)+n-1. (End) LINKS FORMULA a(n) = (1/n^2)*Sum_{d|n} d*phi(d)*n^(n/d). - Vladeta Jovovic, Aug 14 2007, Aug 24 2007 EXAMPLE The 5 necklaces for n=3 are: 000, 001, 002, 012 and 021. MATHEMATICA tor8={}; ru8=Thread[ i_ ->Table[ Mod[i+k, 8], {k, 8}]]; Do[idi=IntegerDigits[k, 8, 8]; try= Function[w, First[temp=Union[Join @@(Table[RotateRight[w, k], {k, 8}]/.#&)/@ ru8]]][idi]; If[idi===try, tor8=Flatten[ {tor8, {{Length[temp], idi}}}, 1] ], {k, 0, 8^8-1}]; PROG (PARI) a(n) = sumdiv(n, d, d*eulerphi(d)*n^(n/d))/n^2; \\ Michel Marcus, Aug 05 2016 CROSSREFS Cf. A002075-A002076. Cf. A081720. Cf. A000312 All endofunctions Cf. A000169 Classes under translation mod n Cf. A001700 Classes under sort Cf. A056665 Classes under rotation Cf. A168658 Classes under complement to n+1 Cf. A130293 Classes under translation and rotation Cf. A081721 Classes under rotation and reversal Cf. A275549 Classes under reversal Cf. A275550 Classes under reversal and complement Cf. A275551 Classes under translation and reversal Cf. A275552 Classes under translation and complement Cf. A275553 Classes under translation, complement and reversal Cf. A275554 Classes under translation, rotation and complement Cf. A275555 Classes under translation, rotation and reversal Cf. A275556 Classes under translation, rotation, complement and reversal Cf. A275557 Classes under rotation and complement Cf. A275558 Classes under rotation, complement and reversal Sequence in context: A012321 A012519 A076795 * A156073 A006366 A012317 Adjacent sequences:  A130290 A130291 A130292 * A130294 A130295 A130296 KEYWORD nonn AUTHOR Wouter Meeussen, Aug 06 2007, Aug 14 2007 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 17 18:48 EST 2019. Contains 319251 sequences. (Running on oeis4.)