A129526 Number of n-bead two-color bracelets with 00 prohibited.

2, 2, 3, 3, 5, 5, 8, 9, 14, 16, 26, 31, 49, 64, 99, 133, 209, 291, 455, 657, 1022, 1510, 2359, 3545, 5536, 8442, 13201, 20319, 31836, 49353, 77436, 120711, 189674, 296854, 467160, 733363, 1155647, 1818594, 2869378, 4524081, 7146483

Offset: 2

Author

Colin Mallows, May 29 2007

Keywords

• nonn

Comments

Bracelets can be turned over; turning the seventh example gives a different necklace but the same bracelet.

a(n) is also the number of inequivalent compositions of n into parts 1 and 2 where two compositions are considered equivalent if one can be obtained from the other by a cyclic rotation and/or reversing of the summands. a(7) = 5 because we have: 2+2+2+1, 2+2+1+1+1, 2+1+2+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1. - Geoffrey Critzer, Feb 01 2014

a(n) is also the average of sequence A000358(n) and Fib(floor(n/2)+2). The expression (1+x)*(1+x^2)/(1-x^2-x^4) (due to H. Kociemba) is the g.f. of Fib(floor(n/2)+2). Even though the offset of a(n) is set at n = 2, the formula is true even for n=1 because a(1) = 1 = (1+1)/2 (since the sequence 1 on a circle does not allow the pattern 00 when it is allowed to wrap around itself on the circle, while the sequence 0 does). - Petros Hadjicostas, Jan 04 2017

Examples

			a(9) = 9 because of 111111111, 011111111, 010111111, 011011111, 011101111, 010101111, 010110111, 011011011, 010101011.
		

Links

• Ali Reza Ashrafi, Jernej Azarija, Khadijeh Fathalikhani, Sandi Klavžar, Marko Petkovšek, Vertex and edge orbits of Fibonacci and Lucas cubes, arXiv:1407.4962 [math.CO], 2014. See Table 4.
• A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
• M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard and B. M. McCoy, Hard hexagon partition function for complex fugacity, arXiv preprint arXiv:1306.6389 [math-ph], 2013.
• M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard and B. M. McCoy, Integrability vs non-integrability: Hard hexagons and hard squares compared, arXiv preprint 1406.5566 [math-ph], 2014.
• C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, Z. Papić, Quantum scarred eigenstates in a Rydberg atom chain: entanglement, breakdown of thermalization, and stability to perturbations, arXiv:1806.10933 [cond-mat.quant-gas], 2018.

Crossrefs

Cf. A000358.

Programs

Mathematica

nn=48;Drop[Map[Total,Transpose[Map[PadRight[#,nn]&,Table[CoefficientList[Series[CycleIndex[DihedralGroup[n],s]/.Table[s[i]->x^i+x^(2i),{i,1,n}],{x,0,nn}],x],{n,0,nn}]]]],2] (* Geoffrey Critzer, Feb 01 2014 *)
mx:=50;CoefficientList[Series[(Sum[(EulerPhi[n] Log[1- x^n (1+x^n)])/n,{n,1,mx}]+((1+x) (1+x^2))/(-1+x^2+x^4))/(-2),{x,0,mx}],x]  (* Herbert Kociemba, Dec 04 2016 *)

Formula

G.f.: [Sum_{n>=1} phi(n)*log(1- x^n*(1+x^n))/n + ((1+x)*(1+x^2))/(-1+x^2+x^4)]/(-2). - Herbert Kociemba, Dec 04 2016

a(n) = [A000358(n)+Fib(floor(n/2)+2)]/2. - Petros Hadjicostas, Jan 04 2017

a(n) = [Fib(floor(n/2)+2)+(1/n) * sum_{d divides n} phi(n/d)*(Fib(d-1)+Fib(d+1))]/2. - Petros Hadjicostas, Jan 04 2017 (with help from Lingyun Zhang).

Extensions

a(10) corrected and added more terms (from a(14) inclusive) by Washington Bomfim, Aug 24 2008