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

Author

N. J. A. Sloane, Mar 25 2017

Keywords

• nonn
• tabf

Examples

			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,
  ...
		

Links

• 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].

Crossrefs

Cf. A094686, A283834-A283837.

For row sums see A283839.

Programs

Mathematica

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 *)