%I A120643
%S A120643 1,1,1,2,1,1,3,2,2,1,5,4,3,3,1,8,8,5,6,4,1,14,14,10,10,10,5,1,24,25,21,
%T A120643 16,20,15,6,1,43,43,43,28,35,35,21,7,1,77,76,83,56,57,70,56,28,8,1
%N A120643 Table T(n,k) = number of fractal initial sequences (where new values 
               are successive integers) of length n whose last term is k.
%C A120643 A fractal sequence is one where, when the first instance of each integer 
               is removed, the original sequence results. We require also that these 
               first instances occur in order: 1,1,2,3 is OK, but 1,1,3,2 is not. 
               A finite sequence is an initial subsequence of (uncountably many) 
               fractal sequences when the result after removing the first instance 
               of each number is an initial subsequence. The total number of such 
               sequences of length n is 2^{n-1}. At each index after the first, 
               the next value can be either a new value or a uniquely determined 
               repetition of some earlier value. Conjecture: column 1 of this array 
               is A007059.
%H A120643 C. Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">
               Fractal sequences</a>
%F A120643 If 2 <= n <= 2k-1, T(n,k) = C(n-2,k-2).
%e A120643 For n = 3, the 4 sequences are 1,1,1; 1,1,2; 1,2,1; and 1,2,3. Of these, 
               2 end in 1, 1 in 2 and 1 in 3, so row 3 is 2,1,1.
%e A120643 The table starts:
%e A120643 1
%e A120643 1,1
%e A120643 2,1,1
%e A120643 3,2,2,1
%e A120643 5,4,3,3,1
%e A120643 8,8,5,6,4,1
%Y A120643 Cf: A007059.
%Y A120643 Sequence in context: A029253 A016441 A131333 this_sequence A111867 A133776 
               A060118
%Y A120643 Adjacent sequences: A120640 A120641 A120642 this_sequence A120644 A120645 
               A120646
%K A120643 more,nonn,tabl
%O A120643 1,4
%A A120643 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 17 2006