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%I M2346 #178 Sep 03 2023 10:42:29
%S 1,1,3,4,8,10,16,20,29,35,47,56,72,84,104,120,145,165,195,220,256,286,
%T 328,364,413,455,511,560,624,680,752,816,897,969,1059,1140,1240,1330,
%U 1440,1540,1661,1771,1903,2024,2168,2300,2456,2600,2769,2925,3107,3276
%N Expansion of (1-x+x^2)/((1-x)^2*(1-x^2)*(1-x^4)).
%C Also number of n X 2 binary matrices under row and column permutations and column complementations (if offset is 0).
%C Also Molien series for certain 4-D representation of dihedral group of order 8.
%C With offset 4, number of bracelets (turnover necklaces) of n-bead of 2 colors with 4 red beads. - _Washington Bomfim_, Aug 27 2008
%C From _Vladimir Shevelev_, Apr 23 2011: (Start)
%C Also number of non-equivalent necklaces of 4 beads each of them painted by one of n colors.
%C The sequence solves the so-called Reis problem about convex k-gons in case k=4 (see our comment to A032279). (End)
%C Number of 2 X 2 matrices with nonnegative integer values totaling n under row and column permutations. - _Gabriel Burns_, Nov 08 2016
%C From _Petros Hadjicostas_, Jan 12 2019: (Start)
%C By "necklace", _Vladimir Shevelev_ (above) means "turnover necklace", i.e., a bracelet. Zagaglia Salvi (1999) also uses this terminology: she calls a bracelet "necklace" and a necklace "cycle".
%C According to Cyvin et al. (1997), the sequence (a(n): n >= 0) consists of "the total numbers of isomers of polycyclic conjugated hydrocarbons with q + 1 rings and q internal carbons in one ring (class Q_q)", where q = 4 and n is the hydrogen content (i.e., we count certain isomers of C_{n+2*q} H_n with q = 4 and n >= 0). (End)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.
%H T. D. Noe, <a href="/A005232/b005232.txt">Table of n, a(n) for n = 0..1000</a>
%H Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi <a href="http://ftp.math.uni-rostock.de/pub/romako/heft68/bouroubi68.html">Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon.</a> Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 1.
%H S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, <a href="http://zfn.mpdl.mpg.de/data/Reihe_A/52/ZNA-1997-52a-0867.pdf">Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem</a>, Z. Naturforsch., 52a (1997), 867-873.
%H S. N. Ethier and S. E. Hodge, <a href="http://dx.doi.org/10.1002/ajmg.1320220207">Identity-by-descent analysis of sibship configurations</a>, Amer. J. Medical Genetics, 22 (1985), 263-272.
%H H. Gupta, <a href="https://web.archive.org/web/20200806162943/https://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a66_964.pdf">Enumeration of incongruent cyclic k-gons</a>, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
%H W. D. Hoskins and Anne Penfold Street, <a href="http://dx.doi.org/10.1017/S1446788700017547">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
%H W. D. Hoskins and A. P. Street, <a href="/A005513/a005513_1.pdf">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
%H M. Klemm, <a href="http://dx.doi.org/10.1007/BF01198572">Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4</a>, Arch. Math. (Basel), 53 (1989), 201-207.
%H P. Lisonek, <a href="/A005045/a005045_2.pdf">Quasi-polynomials: A case study in experimental combinatorics</a>, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H Vladimir Shevelev, <a href="http://www.math.bgu.ac.il/~shevelev/Shevelev_Neclaces.pdf">Necklaces and convex k-gons</a>, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/0710.1370">A problem of enumeration of two-color bracelets with several variations</a>, arXiv:0710.1370 [math.CO], 2007-2011.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1104.4051">Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma)</a> (Cf. Section 5), arXiv:1104.4051 [math.CO], 2011.
%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,2,-2,0,2,-1).
%F G.f.: (1+x^3)/((1-x)*(1-x^2)^2*(1-x^4)).
%F G.f.: (1/8)*(1/(1-x)^4+3/(1-x^2)^2+2/(1-x)^2/(1-x^2)+2/(1-x^4)). - _Vladeta Jovovic_, Aug 05 2000
%F Euler transform of length 6 sequence [ 1, 2, 1, 1, 0, -1 ]. - _Michael Somos_, Feb 01 2007
%F a(2n+1) = A006918(2n+2)/2;
%F a(2n) = (A006918(2n+1) + A008619(n))/2.
%F a(n) = -a(-6 - n) for all n in Z. - _Michael Somos_, Feb 05 2011
%F From _Vladimir Shevelev_, Apr 22 2011: (Start)
%F if n == 0 (mod 4), then a(n) = n*(n^2-3*n+8)/48;
%F if n == 1, 3 (mod 4), then a(n) = (n^2-1)*(n-3)/48;
%F if n == 2 (mod 4), then a(n) = (n-2)*(n^2-n+6)/48. (End)
%F a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - 2*a(n-5) + 2*a(n-7) - a(n-8) with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 8, a(5) = 10, a(6) = 16, a(7) = 20. - _Harvey P. Dale_, Oct 24 2012
%F a(n) = ((n+3)*(2*n^2+12*n+19+9*(-1)^n) + 6*(-1)^((2*n-1+(-1)^n)/4)*(1+(-1)^n))/96. - _Luce ETIENNE_, Mar 16 2015
%F a(n) = |A128498(n)| + |A128498(n-3)|. - _R. J. Mathar_, Jun 11 2019
%e G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 10*x^5 + 16*x^6 + 20*x^7 + 29*x^8 + ...
%e There are 8 4 X 2 matrices up to row and column permutations and column complementations:
%e [1 1] [1 0] [1 0] [0 1] [0 1] [0 1] [0 1] [0 0]
%e [1 1] [1 1] [1 0] [1 0] [1 0] [1 0] [0 1] [0 1]
%e [1 1] [1 1] [1 1] [1 1] [1 0] [1 0] [1 0] [1 0]
%e [1 1] [1 1] [1 1] [1 1] [1 1] [1 0] [1 0] [1 1].
%e There are 8 2 X 2 matrices of nonnegative integers totaling 4 up to row and column permutations:
%e [4 0] [3 1] [2 2] [2 1] [2 1] [3 0] [2 0] [1 1]
%e [0 0] [0 0] [0 0] [0 1] [1 0] [1 0] [2 0] [1 1].
%p A005232:=-(-1-z-2*z**3+2*z**2+z**7-2*z**6+2*z**4)/(z**2+1)/(1+z)**2/(z-1)**4; # conjectured by _Simon Plouffe_ in his 1992 dissertation; gives sequence apart from an initial 1
%t k = 4; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* _Robert A. Russell_, Sep 27 2004 *)
%t CoefficientList[ Series[(1 - x + x^2)/((1 - x)^2(1 - x^2)(1 - x^4)), {x, 0, 51}], x] (* _Robert G. Wilson v_, Mar 29 2006 *)
%t LinearRecurrence[{2,0,-2,2,-2,0,2,-1},{1,1,3,4,8,10,16,20},60] (* _Harvey P. Dale_, Oct 24 2012 *)
%t k=4 (* Number of red beads in bracelet problem *); CoefficientList[Series[(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2,{x,0,50}],x] (* _Herbert Kociemba_, Nov 04 2016 *)
%o (PARI) {a(n) = (n^3 + 9*n^2 + (32-9*(n%2))*n + [48, 15, 36, 15][n%4+1]) / 48}; \\ _Michael Somos_, Feb 01 2007
%o (PARI) {a(n) = my(s=1); if( n<-5, n = -6-n; s=-1); if( n<0, 0, s * polcoeff( (1 - x + x^2) / ((1 - x)^2 * (1 - x^2) * (1 - x^4)) + x * O(x^n), n))}; \\ _Michael Somos_, Feb 01 2007
%o (PARI) a(n) = round((n^3 +9*n^2 +(32-9*(n%2))*n)/48 +0.6) \\ _Washington Bomfim_, Jul 17 2008
%o (PARI) a(n) = ceil((n+1)*(2*n^2+16*n+39+9*(-1)^n)/96) \\ _Tani Akinari_, Aug 23 2013
%o (Python) a=lambda n: sum((k//2+1)*((n-k)//2+1) for k in range((n-1)//2+1))+(n+1)%2*(((n//4+1)*(n//4+2))//2) # _Gabriel Burns_, Nov 08 2016
%Y Row n=2 of A343875.
%Y Column k=4 of A052307.
%Y Cf. A006381, A006382, A008805.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Sequence extended by _Christian G. Bower_
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