A242510 Number of n-length words on {1,2,3} such that the maximal blocks (runs) of 1's have odd length, the maximal blocks of 2's have even length and the maximal blocks of 3's have odd length.
1, 2, 3, 8, 15, 32, 67, 138, 289, 600, 1249, 2600, 5409, 11258, 23427, 48752, 101455, 211128, 439363, 914322, 1902721, 3959600, 8240001, 17147600, 35684481, 74260082, 154536643, 321593688, 669242575, 1392706512, 2898248707
Offset: 0
Keywords
Examples
a(3)=8 because we have: 111, 122, 131, 221, 223, 313, 322, 333.
Links
- Index entries for linear recurrences with constant coefficients, signature (1,2,1,-1).
Programs
-
Mathematica
n=3;nn=30;CoefficientList[Series[1/(1-Sum[v[i]/(1+v[i]),{i,1,n}])/.Join[Table[v[i]->z/(1-z^2),{i,1,n,2}],Table[v[i]->z^2/(1-z^2),{i,2,n,2}]],{z,0,nn}],z] (* Changing n=3 at the beginning of this code to n = k, (for k a positive integer) will return the number of n-length words on {1,2,...,k} where the maximal run lengths of odd integers are odd and the maximal run lengths of even integers are even. *)
Formula
G.f.: (1 + x - x^2)/(1 - x - 2*x^2 - x^3 +x^4).
a(n) = a(n-1) +2*a(n-2) +a(n-3) -a(n-4). - Fung Lam, May 18 2014