A136704 -id:A136704 - cp's OEIS frontend

A253076 Bisection of A136704.

1, 5, 30, 205, 1476, 11070, 85410, 672605, 5380830, 43584804, 356602950, 2941974270, 24441017580, 204257075490, 1715759433624, 14476720225405, 122626336026960, 1042323856225470, 8887182353111790, 75985409119105764, 651303506735164140, 5595289216952336550

Offset: 1

N. J. A. Sloane, Feb 01 2015

nonn

A sequence in Table 1 of Hao (1991) appears to match this sequence, but there are not enough terms there to be certain. It is possible that Hao's 1989 book will clarify things, but I do not have access to it.

Hao, Bai Lin, Elementary symbolic dynamics and chaos in dissipative systems. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. xvi+460 pp. ISBN: 9971-50-682-3; 9971-50-698-X

Andrew Howroyd, Table of n, a(n) for n = 1..200
Bai-lin Hao, Symbolic dynamics and characterization of complexity, Physica, D51 (1991), 161-176.

Cf. A136704, A253077.

a[n_] := Sum[Boole[OddQ[d]] MoebiusMu[d] (3^(2n/d)-1), {d, Divisors[n]}]/(8n);
Array[a, 22] (* Jean-François Alcover, Aug 26 2019 *)

a(n) = sumdiv(n>>valuation(n,2), d, moebius(d)*(3^(2*n/d)-1))/(8*n); \\ Andrew Howroyd, Nov 11 2018

a(n) = (1/8*n) * Sum_{d|n, d odd} mu(d)*(3^(2*n/d) - 1). - Andrew Howroyd, Nov 11 2018

Offset corrected and terms a(14) and beyond from Andrew Howroyd, Nov 11 2018

A253077 Bisection of A136704 (divided by 2).

Original entry on oeis.org

0, 1, 6, 39, 273, 2013, 15330, 119572, 949560, 7646457, 62263994, 511647711, 4236443046, 35303691969, 295820592090, 2490618533403, 21057047599918, 178684089639276, 1521229411793910, 12988958823773660, 111198159686496300, 954235370332956621, 8206424184863387028

Offset: 1

N. J. A. Sloane, Feb 01 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A136704, A253076.

Offset corrected and more terms added by Amiram Eldar, May 08 2024

A136703 Number of Lyndon words on {1,2,3} with an even number of 1's and an even number of 2's.

Original entry on oeis.org

1, 0, 2, 3, 12, 26, 78, 195, 546, 1452, 4026, 11010, 30660, 85254, 239144, 672195, 1899120, 5379738, 15292914, 43581852, 124527988, 356594898, 1023295422, 2941952130, 8472886092, 24440956260, 70607383938

Offset: 1

Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Jan 16 2008

nonn

A Lyndon word is the aperiodic necklace representative which is lexicographically earliest among its cyclic shifts. Thus we can apply the fixed density formulas: L_k(n,d)=sum L(n-d, n_1,..., n_(k-1)); n_1+...+n_(k-1)=d where L(n_0, n_1,...,n_(k-1))=(1/n)sum mu(j)*[(n/j)!/((n_0/j)!(n_1/j)!...(n_(k-1)/j)!)]; j|gcd(n_0, n_1,...,n_(k-1)). For this sequence, sum over n_0,n_1=even.

			For n=3, out of 8 possible Lyndon words: 112, 113, 122, 123, 132, 133, 223, 233, only 113 and 223 have an even number of both 1's and 2's. Thus a(3)=2.

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
F. Ruskey and J. Sawada, An Efficient Algorithm for Generating Necklaces with Fixed Density, SIAM J. Computing, 29 (1999) 671-684.
M. Zabrocki, MATH5020 York University Course Website

Cf. A006575; A027376; A133267; A136704.

a(1)=1; for n>1, if n=odd then a(n)= sum(mu(d)*3^(n/d))/(4n); d|n. If n=even, then a(n)= sum(mu(d)*3^(n/d))/n; d|n -(3/4)*sum(mu(d)*(3^(n/d)-1))/n; d|n, d odd.

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A253076 Bisection of A136704.

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A253077 Bisection of A136704 (divided by 2).

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A136703 Number of Lyndon words on {1,2,3} with an even number of 1's and an even number of 2's.

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