%I
%S 1,2,5,1,10,1,2,21,1,2,5,1,42,1,2,5,1,10,1,2,85,1,2,5,1,10,1,2,21,1,2,
%T 5,1,170,1,2,5,1,10,1,2,21,1,2,5,1,42,1,2,5,1,10,1,2,341,1,2,5,1,10,1,
%U 2,21,1,2,5,1,42,1,2,5,1,10,1,2,85,1,2,5,1,10,1,2,21,1,2,5,1,682,1,2,5,1
%N Runs of zeros in binomial(3k,k)/(2k+1) (Mod 2): relates ternary trees (A001764) to the infinite Fibonacci word (A003849).
%C Has complementary parity to the infinite Fibonacci word: a(n) = 1  A003849(n) (Mod 2). Records are given by A000975 and occur at Fibonacci numbers: {1,2,5,10,21,42,85,...} occur at {1,2,3,5,8,13,21,...}.
%F Construction: start with strings S(1)={1} and S(2)={1, 2}; for k>2, let L=largest number in current string S(k); to obtain S(k+1), append S(k1) to the end of S(k) and then replace the last number in this resulting string with {2L+1 (k odd) or 2L (k even)}. String lengths have Fibonacci growth: {1}, {1, 2}, {1, 2, 5}, {1, 2, 5, 1, 10}, {1, 2, 5, 1, 10, 1, 2, 21}, ...
%Y Cf. A001764 (ternary trees), A003849 (infinite Fibonacci word), A000975 (records), A085357.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 25 2003
