A085358 Runs of zeros in binomial(3k,k)/(2k+1) (Mod 2): relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

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, 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, 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

Offset: 0

Author

Paul D. Hanna, Jun 25 2003

Keywords

• nonn

Comments

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,...}.

Crossrefs

Cf. A001764 (ternary trees), A003849 (infinite Fibonacci word), A000975 (records), A085357.

Formula

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(k-1) 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}, ...