A176354 The number of words of length n created with the letters a, b, c with at least as many a's as b's and at least as many b's as c's and no adjacent letters forming the pattern aba.

1, 1, 3, 9, 17, 46, 153, 338, 935, 3067, 7182, 20064, 64819, 156968, 441084, 1408612, 3488489, 9840608, 31145791, 78388512, 221722954, 696757333, 1775204952, 5031253931, 15718503073, 40435757688, 114779701466, 356852697226, 925209496348, 2629513384131

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Maple

a:= n-> add(add(w(na, nb, n-na-nb, 0),
        nb=ceil((n-na)/2)..min(n-na, na)), na=ceil(n/3)..n):
w:= proc(a, b, c, x) option remember;
      `if`([a, b, c]=[0$3], 1, `if`(a>0 and x<2, w(a-1, b, c, 1), 0)+
      `if`(b>0, w(a, b-1, c, `if`(x=1, 2, 0)), 0)+
      `if`(c>0, w(a, b, c-1, 0), 0))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2012

Mathematica

a[n_] := Sum[Sum[w[na, nb, n - na - nb, 0], {nb, Ceiling[(n - na)/2], Min[n - na, na]}], {na, Ceiling[n/3], n}];
w[a_, b_, c_, x_] := w[a, b, c, x] = If[{a, b, c} == {0, 0, 0}, 1, If[a > 0 && x < 2, w[a - 1, b, c, 1], 0] + If[b > 0, w[a, b - 1, c, If[x == 1, 2, 0]], 0] + If[c > 0, w[a, b, c - 1, 0], 0]];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)

Prog

(Sage)
def atleastcond(w):
    return w.count(0) >= w.count(1) >= w.count(2)
def avoidsaba(w):
    for i in range(len(w)-2):
        if w[i] == 0 and w[i+1] == 1 and w[i+2] == 0:
            return False
    return True
len([w for w in Words([0, 1, 2], length=6) if atleastcond(w) and avoidsaba(w)])
# Jenny Li (jiemi2007(AT)yahoo.ca), Apr 17 2010

Crossrefs

Cf. A176148, A206701.

Sequence in context: A348382 A176148 A206701 * A210337 A173140 A018307

Adjacent sequences: A176351 A176352 A176353 * A176355 A176356 A176357

Keyword

nonn

Author

Jenny Li (jiemi2007(AT)yahoo.ca), Apr 15 2010, Apr 17 2010

Extensions

Extended beyond a(15) by Alois P. Heinz, May 22 2012

Status

approved