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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.

%I A260697 #15 Mar 02 2022 11:54:31
%S A260697 1,2,4,6,11,18,32,54,95,164,291,514,923,1656,3000,5442,9942,18216,
%T A260697 33564,62040,115167,214404,400497,750070,1408734,2652088,5004833,
%U A260697 9464616,17935137,34049044,64754844,123351410,235335966,449632300,860241606,1647932000
%N 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.
%H A260697 Alois P. Heinz, <a href="/A260697/b260697.txt">Table of n, a(n) for n = 0..1000</a>
%e A260697 a(3) = 6: 000, 001, 011, 100, 110, 111.
%e A260697 a(4) = 11: 0000, 0001, 0011, 0110, 0111, 1000, 1001, 1010, 1100, 1110, 1111.
%e A260697 a(5) = 18: 00000, 00001, 00011, 00110, 00111, 01100, 01110, 01111, 10000, 10001, 10011, 10100, 11000, 11001, 11010, 11100, 11110, 11111.
%e A260697 a(10) = 291: 0000000000, 0000000001, 0000000011, ..., 0110101010, 1010101000, 1010101001, 1010101010, 1101010100, 1110101010, ..., 1111111100, 1111111110, 1111111111.
%p A260697 b:= proc(n, t, c) option remember;
%p A260697      `if`(c<0, 0, `if`(n=0, `if`(c=0, 1, 0),
%p A260697       b(n-1, [2, 4, 6, 4, 6, 4, 6][t], c-`if`(t=5, 1, 0))+
%p A260697       b(n-1, [3, 5, 7, 5, 7, 5, 7][t], c+`if`(t=6, 1, 0))))
%p A260697     end:
%p A260697 a:= n-> b(n, 1, 0):
%p A260697 seq(a(n), n=0..40);
%p A260697 # second Maple program:
%p A260697 a:= proc(n) option remember;
%p A260697      `if`(n<7, [1, 2, 4, 6, 11, 18, 32][n+1],
%p A260697      ((n+3)*(307*n^2-2357*n+196)              *a(n-1)
%p A260697       -(19280-3372*n-5181*n^2+719*n^3)        *a(n-2)
%p A260697       +(2*(6582+268*n^3-2857*n^2+6959*n))     *a(n-3)
%p A260697       +(2*(-3307*n^2+1151*n+384*n^3+9052))    *a(n-4)
%p A260697       -(2*(1016*n^3-12133*n^2+38927*n-28304)) *a(n-5)
%p A260697       +(4*(27387*n+431*n^3-38420-6108*n^2))   *a(n-6)
%p A260697       -(4*(n-7))*(67*n-236)*(2*n-11)          *a(n-7)
%p A260697       )/((2*(n+4))*(24*n^2-148*n-279)))
%p A260697     end:
%p A260697 seq(a(n), n=0..40);
%t A260697 b[n_, t_, c_] := b[n, t, c] =
%t A260697      If[c < 0, 0, If[n == 0, If[c == 0, 1, 0],
%t A260697      b[n - 1, {2, 4, 6, 4, 6, 4, 6}[[t]], c - If[t == 5, 1, 0]] +
%t A260697      b[n - 1, {3, 5, 7, 5, 7, 5, 7}[[t]], c + If[t == 6, 1, 0]]]];
%t A260697 a[n_] := b[n, 1, 0];
%t A260697 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 02 2022, after _Alois P. Heinz_ *)
%Y A260697 Cf. A118430, A164146, A255386, A260505, A260668.
%K A260697 nonn
%O A260697 0,2
%A A260697 _Alois P. Heinz_, Nov 16 2015