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A366043
Number of circular binary sequences of length n with an odd number of 0's and no consecutive 1's.
2
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
1,2
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.
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.
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).
EXAMPLE
For n = 5, the a(5) = 6 allowed sequences are 00000, 00101, 01001, 01010, 10010, 10100.
MATHEMATICA
LinearRecurrence[{0, 1, 2, 1}, {0, 1, 2, 1}, 50]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Joshua P. Bowman, Sep 27 2023
STATUS
approved