A366043 Number of circular binary sequences of length n with an odd number of 0's and no consecutive 1's.

1, 2, 1, 4, 6, 8, 15, 24, 37, 62, 100, 160, 261, 422, 681, 1104, 1786, 2888, 4675, 7564, 12237, 19802, 32040, 51840, 83881, 135722, 219601, 355324, 574926, 930248, 1505175, 2435424, 3940597, 6376022, 10316620, 16692640, 27009261, 43701902, 70711161, 114413064, 185124226, 299537288, 484661515, 784198804, 1268860317

Offset: 1

Author

Joshua P. Bowman, Sep 27 2023

Keywords

• nonn
• easy

Comments

A circular binary sequence is a finite sequence of 0's and 1's for which the first and last digits are considered to be adjacent. Rotations are distinguished from each other. Also called a marked cyclic binary sequence.

a(n) is also equal to the number of matchings in the cycle graph C_n for which the number of edges plus the number of unmatched vertices is odd.

a(n) is also equal to the number of circular compositions of n into an odd number of 1's and 2's.

Examples

			For n = 5, the a(5) = 6 allowed sequences are 00000, 00101, 01001, 01010, 10010, 10100.
		

Links

• Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 19.
• Petros Hadjicostas and Lingyun Zhang, On cyclic strings avoiding a pattern, Discrete Mathematics, 341 (2018), 1662-1674.
• W. O. J. Moser, Cyclic binary strings without long runs of like (alternating) bits, Fibonacci Quart. 31 (1993), no. 1, 2-6.
• Index entries for linear recurrences with constant coefficients, signature (0,1,2,1).

Crossrefs

Cf. A000204, A061347, A100886.

Programs

Mathematica

LinearRecurrence[{0, 1, 2, 1}, {0, 1, 2, 1}, 50]

Formula

G.f.: x*(1+2*x)/((1-x-x^2)*(1+x+x^2)).

a(n) = a(n-2) + 2*a(n-3) + a(n-4), a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 1.

a(n) = (A000204(n) + A061347(n))/2.

a(n) = (1/2)*A000204(n) - cos(2*Pi*n/3).

a(n) = A000204(n) - A100886(n-1).