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Revision History for A066313

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A066313

Number of aperiodic necklaces with n red or blue beads such that two necklaces are equivalent under the operation of simultaneously turning the necklace over and switching the two colors.
(history; published version)

#11 by N. J. A. Sloane at Wed Jun 16 15:35:21 EDT 2021

STATUS

reviewed

approved

#10 by Joerg Arndt at Wed Jun 16 02:07:59 EDT 2021

STATUS

proposed

reviewed

Discussion
Wed Jun 16	08:42	Andrew Howroyd: sure

#9 by Andrew Howroyd at Tue Jun 15 22:39:33 EDT 2021

STATUS

editing

proposed

Discussion
Wed Jun 16	01:52	Michel Marcus: so can you add pari for/in A053656

#8 by Andrew Howroyd at Mon Jun 14 18:43:29 EDT 2021

	NAME	`Number of aperiodic necklaces with n red or blue beads such that two necklaces are equivalent under the operation (turn of simultaneously turning the necklace over and switchswitching the two colors)..`
	DATA	`1, 1, 1, 2, 3, 6, 9, 18, 28, 57, 93, 181, 315, 612, 1091, 2100, 3855, 7392, 13797, 26436, 49929, 95790, 182361, 350440, 671088, 1292445, 2485504, 4797261, 9256395, 17903316, 34636833, 67124160, 130150493, 252675975, 490853403, 954498874, 1857283155, 3616938738`
	LINKS	`Andrew Howroyd, <a href="/A066313/b066313.txt">Table of n, a(n) for n = 1..1000</a>`
	PROG	`(PARI) \\ here b(n) is A053656.` `b(n)={(sumdiv(n, d, eulerphi(d)2^(n/d))/n + if(n%2==0, 2^(n/2-1)))/2}` `a(n)={sumdiv(n, d, moebius(d)b(n/d))} \\ Andrew Howroyd, Jun 14 2021`
	CROSSREFS	`Cf. A053656.`
	EXTENSIONS	`Terms a(36) and beyond from Andrew Howroyd, Jun 14 2021`
	STATUS	`approved` `editing`

#7 by Russ Cox at Fri Mar 30 17:37:12 EDT 2012

AUTHOR

_Christian G. Bower (bowerc(AT)usa.net), _, Dec 13 2001; revised Apr 25 2006

Discussion
Fri Mar 30	17:37	OEIS Server: https://oeis.org/edit/global/178

#6 by Russ Cox at Sun Jul 10 18:19:50 EDT 2011

LINKS

<a href="/Sindx_index/Br.html#bracelets">Index entries for sequences related to bracelets</a>

Discussion
Sun Jul 10	18:19	OEIS Server: https://oeis.org/edit/global/25

#5 by N. J. A. Sloane at Thu Nov 11 07:34:06 EST 2010

LINKS

N. J. A. Sloane, <a href="="/transforms.txt">Transforms</a>

<a href="="/Sindx_Br.html#bracelets">Index entries for sequences related to bracelets</a>

KEYWORD

nonn,new

nonn

#4 by N. J. A. Sloane at Fri Feb 27 03:00:00 EST 2009

LINKS

N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>

<a href="http://www.research.att.com/~njas/sequences/Sindx_Br.html#bracelets">Index entries for sequences related to bracelets</a>

KEYWORD

nonn,new

nonn

#3 by N. J. A. Sloane at Fri Feb 24 03:00:00 EST 2006

	NAME	`Number of aperiodic bracelets (or necklaces) with n red or blue beads such that two necklaces are equivalent under the beadsoperation (turn switchnecklace colorsover whenand braceletswitch isthe turnedtwo over.colors).`
	COMMENTS	`Also number of aperiodic cyclic graphs with oriented edges on n nodes that can be turned over.`
	LINKS	`<a href="http://www.research.att.com/~njas/sequences/Sindx_Br.html#bracelets">Index entries for sequences related to bracelets</a>` `<a href="http://www.research.att.com/~njas/sequences/Sindx_Br.html#bracelets">Index entries for sequences related to bracelets</a>`
	EXAMPLE	`The equivalence requires the "turning over" operation and the "switching colors" operation to be simultaneous; thus rrrbbrrb is equivalent to rbbrrbbb, but not to bbbrrbbr.`
	AUTHOR	`Christian G. Bower (bowerc(AT)usa.net), Dec 13 2001; revised Apr 25 2006`

#2 by N. J. A. Sloane at Wed Sep 22 03:00:00 EDT 2004

NAME

AperiodicNumber of aperiodic bracelets (or necklaces) with n red or blue beads such that the beads switch colors when bracelet is turned over.

KEYWORD

nonn,new

nonn

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Last modified August 18 23:05 EDT 2024. Contains 375284 sequences. (Running on oeis4.)