A283834 -id:A283834 - cp's OEIS frontend

A283838 Irregular triangle read by rows: T(n,k) (n >= 8, 3 <= k <= floor(n/2)-1) = number of binary vectors of length <= n that start with 1^k, 0, end with 1, 0^k, and the factor between 1^k and 0^k does not contain 0^k or 1^k.

1, 3, 5, 1, 9, 3, 16, 7, 1, 26, 13, 3, 43, 25, 7, 1, 71, 47, 15, 3, 115, 88, 29, 7, 1, 187, 162, 57, 15, 3, 304, 299, 111, 31, 7, 1, 492, 551, 215, 61, 15, 3, 797, 1015, 416, 121, 31, 7, 1, 1291, 1867, 802, 239, 63, 15, 3, 2089, 3435, 1547, 471, 125, 31, 7, 1, 3381, 6319, 2983, 927, 249, 63, 15, 3

Offset: 8

N. J. A. Sloane, Mar 25 2017

			Triangle begins:
     1,
     3,
     5,     1,
     9,     3,
    16,     7,    1,
    26,    13,    3,
    43,    25,    7,    1,
    71,    47,   15,    3,
   115,    88,   29,    7,   1,
   187,   162,   57,   15,   3,
   304,   299,  111,   31,   7,   1,
   492,   551,  215,   61,  15,   3,
   797,  1015,  416,  121,  31,   7,  1,
  1291,  1867,  802,  239,  63,  15,  3,
  2089,  3435, 1547,  471, 125,  31,  7, 1,
  3381,  6319, 2983,  927, 249,  63, 15, 3,
  5472, 11624, 5751, 1824, 495, 127, 31, 7, 1,
  ...

Alois P. Heinz, Rows n = 8..290, flattened
Stefano Bilotta, Variable-length Non-overlapping Codes, arXiv preprint arXiv:1605.03785 [cs.IT], 2016 [See Table 3].

Cf. A094686, A283834-A283837.

For row sums see A283839.

gf[k_] := x^(2k)(x-x^k)^2 / ((1-x)(1-x^k)(1-2x+x^k));
T[n_, k_] := SeriesCoefficient[gf[k], {x, 0, n}];
Table[T[n, k], {n, 8, 24}, {k, 3, Floor[n/2]-1}] // Flatten (* Jean-François Alcover, Apr 05 2017 *)

A366045 Number of circular binary sequences of length n with an odd number of 0's and no three consecutive 1's.

Original entry on oeis.org

1, 2, 4, 4, 11, 20, 36, 64, 121, 222, 408, 748, 1379, 2536, 4664, 8576, 15777, 29018, 53372, 98164, 180555, 332092, 610812, 1123456, 2066361, 3800630, 6990448, 12857436, 23648515, 43496400, 80002352, 147147264, 270646017, 497795634, 915588916, 1684030564

Offset: 1

Joshua P. Bowman, Sep 27 2023

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 circular compositions of n into an odd number of 1's, 2's, and 3's.

			a(1)=1 because 0 is the only allowed sequence of length one, and a(2)=2 because 01 and 10 are the only allowed sequences of length two.
The allowed sequences of length three are 000, 011, 101, and 110. The allowed sequences of length four are 0001, 0010, 0100, and 1000. Thus a(3)=a(4)=4.

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 (0,1,2,3,2,1).

Cf. A001644, A283834, A366043, A366044.

LinearRecurrence[{0, 1, 2, 3, 2, 1}, {1, 2, 4, 4, 11, 20}, 50]

G.f.: x*(1+2*x+3*x^2)/((1-x-x^2-x^3)*(1+x+x^2+x^3)).
a(n) = (1/2)*A001644(n) + 1/2 - 2*[n==0 (mod 4)].
a(n) = a(n-2) + 2*a(n-3) + 3*a(n-4) + 2*a(n-5) + a(n-6), a(1)=1, a(2)=2, a(3)=4, a(4)=4, a(5)=11, a(6)=20.
a(n) = A001644(n) - A366044(n).

cp's OEIS Frontend

A283838 Irregular triangle read by rows: T(n,k) (n >= 8, 3 <= k <= floor(n/2)-1) = number of binary vectors of length <= n that start with 1^k, 0, end with 1, 0^k, and the factor between 1^k and 0^k does not contain 0^k or 1^k.

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