Thank you, Joshua! I had somehow lived in total ignorance of Newman's amazing, useful, and tiny invention: NestList[1/(2*Floor[#] - # + 1) &, 0, 69]
--Golly, this is amazing. http://oeis.org/A002487 explains, the sequence there is a(0,1,2,3,...) = 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8... Rewriting the recurrence I find a(n+2) = 2*a(n+1)*Floor(a(n)/a(n+1)) + a(n+1) - a(n) for example 7 = 2*4*Floor(5/4) + 4 - 5 = 8+4-5 and 8 = 2*3*Floor(7/3) + 3 - 7 = 12+3-7. They give another recurrence a(2*n) = a(n); a(2*n+1) = a(n)+a(n+1). OEIS claims that L.Carlitz 1964 showed a(n+1) = number of odd binomial(n-k,k) with 0<=k<n/2, but this is just false. Trying to rediscover what the hell this actually should have said, I compute Pascal triangle mod 2: 1 11 101 1111 10001 110011 1010101 11111111 and as Javier Torres pointed out the sums of 45 degree upward-right diagonals are 1,1,2,1,3,2,3,1... so that a(n+1) = sum( binomial(n-k, k) mod 2, k=0..floor(n/2) ) Which shows the following interesting massive bug in MAPLE 9: f := (n) -> sum( binomial(n-k, k) mod 2, k=0..floor(n/2) ); f(34); 9227465 Say WHAT??!! Hence 0 <= a(n) <= (n+1)/2 and it is never 0 after the start.