A111091 Successive generations of a Kolakoski(3,1) rule starting with 1 (see A066983).

1, 3, 111, 313, 1113111, 313111313, 11131113131113111, 3131113131113111313111313, 1113111313111311131311131311131113131113111

Offset: 1

Author

Benoit Cloitre, Oct 12 2005

Keywords

• nonn

Comments

Terms are palindromic. If b_3(n) denotes the number of 3's in a(n) then b(n) satisfies the recursion: b_3(1)=0, b_3(2)=1 and b_3(n) = b_3(n-1) + b_3(n-2) + (-1)^n so that b_3(2n)=A055588(n) and b_3(2n+1)=A027941(n). If b_1(n) denotes the number of 1's: b_1(1)=1, b_1(2)=0 and b_1(n) = b_1(n-1) + b_1(n-2) - 2*(-1)^n so that b_1(2n)=A004146(n) and b_1(2n+1) = A000032(n-2) - 2.

Examples

			1 --> 3 --> 111 --> 313 --> 1113111 --> 313111313
		

Crossrefs

Cf. A111081.

Formula

As n grows, a(2n-1) converges toward A095345 (read as a word) and a(2n) converges toward A095346.