%I
%S 1100,10,1110100,10,11010,10,1111000,10,1100,10,11100,10,
%T 11011111010110111011110100111011011111100111100010101000100,10,
%U 11111010110111011110100111011011111100111100010101000100,10,1100,10,11101100,10,11010,10
%N a(n) = binary odd/even encoding of the iterates in the modified Syracuse algorithm (msa) starting with 2n+1 and continuing up to (but not including) the first iterate less than 2n+1.
%C For the msa mapping see A260590; if x is odd append 1 and if x is even append 0.
%C The binary length of a(n) is A260590(n).
%C For even numbers, 2n, append to f(n) a 0. Example: f(10) = 0, f(5) = 010.
%C Tallying all the ones and zeros, there appear to be five ones for every four zeros.
%C Terms sorted in increasing order and duplicates removed: 10, 1100, 11010, 11100, 1101100, 1110100, 1111000, ...
%C Since msa always starts with an odd number every binary encoding starts with digit 1 and has at least two digits.  _Hartmut F. W. Hoft_, Nov 05 2015
%F a(n) = b_1 b_2 ... b_k, the binary kdigit number where b_j = 1 when the jth iterate of msa is odd and b_j = 0 when it is even, where the first k iterates exceed 2n+1, but the (k+1)st iterate is less than 2n+1.  _Hartmut F. W. Hoft_, Nov 05 2015
%e a(1) = 1100 since A260590(1) is 4, the four operations are, in order following the msa mapping scheme: (3x+1)/2, (3x+1)/2, x/2, and finishing with a x/2 mapping.
%t f[n_] := Block[{k = 2n + 1, lst = {}}, While[k > 2n, If[ OddQ@ k, k = (3k + 1)/2; AppendTo[ lst, 1], k /= 2; AppendTo[ lst, 0]]]; FromDigits@ lst]; Array[f, 22]
%Y Cf. A005408, A176999, A260590.
%K nonn
%O 1,1
%A Joseph K. Horn and _Robert G. Wilson v_, Jul 31 2015
%E Name change by _Hartmut F. W. Hoft_, Nov 05 2015
