A260697 Number of binary words w of length n with equal numbers of 010 and 101 subwords such that for every prefix of w the number of occurrences of subword 101 is larger than or equal to the number of occurrences of subword 010.

1, 2, 4, 6, 11, 18, 32, 54, 95, 164, 291, 514, 923, 1656, 3000, 5442, 9942, 18216, 33564, 62040, 115167, 214404, 400497, 750070, 1408734, 2652088, 5004833, 9464616, 17935137, 34049044, 64754844, 123351410, 235335966, 449632300, 860241606, 1647932000

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Example

a(3) = 6: 000, 001, 011, 100, 110, 111.
a(4) = 11: 0000, 0001, 0011, 0110, 0111, 1000, 1001, 1010, 1100, 1110, 1111.
a(5) = 18: 00000, 00001, 00011, 00110, 00111, 01100, 01110, 01111, 10000, 10001, 10011, 10100, 11000, 11001, 11010, 11100, 11110, 11111.
a(10) = 291: 0000000000, 0000000001, 0000000011, ..., 0110101010, 1010101000, 1010101001, 1010101010, 1101010100, 1110101010, ..., 1111111100, 1111111110, 1111111111.

Maple

b:= proc(n, t, c) option remember;
     `if`(c<0, 0, `if`(n=0, `if`(c=0, 1, 0),
      b(n-1, [2, 4, 6, 4, 6, 4, 6][t], c-`if`(t=5, 1, 0))+
      b(n-1, [3, 5, 7, 5, 7, 5, 7][t], c+`if`(t=6, 1, 0))))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..40);
# second Maple program:
a:= proc(n) option remember;
     `if`(n<7, [1, 2, 4, 6, 11, 18, 32][n+1],
     ((n+3)*(307*n^2-2357*n+196)              *a(n-1)
      -(19280-3372*n-5181*n^2+719*n^3)        *a(n-2)
      +(2*(6582+268*n^3-2857*n^2+6959*n))     *a(n-3)
      +(2*(-3307*n^2+1151*n+384*n^3+9052))    *a(n-4)
      -(2*(1016*n^3-12133*n^2+38927*n-28304)) *a(n-5)
      +(4*(27387*n+431*n^3-38420-6108*n^2))   *a(n-6)
      -(4*(n-7))*(67*n-236)*(2*n-11)          *a(n-7)
      )/((2*(n+4))*(24*n^2-148*n-279)))
    end:
seq(a(n), n=0..40);

Mathematica

b[n_, t_, c_] := b[n, t, c] =
     If[c < 0, 0, If[n == 0, If[c == 0, 1, 0],
     b[n - 1, {2, 4, 6, 4, 6, 4, 6}[[t]], c - If[t == 5, 1, 0]] +
     b[n - 1, {3, 5, 7, 5, 7, 5, 7}[[t]], c + If[t == 6, 1, 0]]]];
a[n_] := b[n, 1, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

Crossrefs

Cf. A118430, A164146, A255386, A260505, A260668.

Sequence in context: A185192 A007053 A005684 * A018167 A295831 A140443

Adjacent sequences: A260694 A260695 A260696 * A260698 A260699 A260700

Keyword

nonn

Author

Alois P. Heinz, Nov 16 2015

Status

approved