|
COMMENTS
|
After expelling 0 and 1, the numbers 4x occupy same positions in S that 1 occupies in the infinite Fibonacci word (A003849).
Numbers k such that the k-th composition in standard order has all odd parts, or numbers k such that A124758(k) is odd. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. For example, the sequence of all compositions into odd parts begins:
0: () 57: (1,1,3,1) 135: (5,1,1,1)
1: (1) 60: (1,1,1,3) 144: (3,5)
3: (1,1) 63: (1,1,1,1,1,1) 147: (3,3,1,1)
4: (3) 64: (7) 153: (3,1,3,1)
7: (1,1,1) 67: (5,1,1) 156: (3,1,1,3)
9: (3,1) 73: (3,3,1) 159: (3,1,1,1,1,1)
12: (1,3) 76: (3,1,3) 192: (1,7)
15: (1,1,1,1) 79: (3,1,1,1,1) 195: (1,5,1,1)
16: (5) 97: (1,5,1) 201: (1,3,3,1)
19: (3,1,1) 100: (1,3,3) 204: (1,3,1,3)
25: (1,3,1) 103: (1,3,1,1,1) 207: (1,3,1,1,1,1)
28: (1,1,3) 112: (1,1,5) 225: (1,1,5,1)
31: (1,1,1,1,1) 115: (1,1,3,1,1) 228: (1,1,3,3)
33: (5,1) 121: (1,1,1,3,1) 231: (1,1,3,1,1,1)
36: (3,3) 124: (1,1,1,1,3) 240: (1,1,1,5)
39: (3,1,1,1) 127: (1,1,1,1,1,1,1) 243: (1,1,1,3,1,1)
48: (1,5) 129: (7,1) 249: (1,1,1,1,3,1)
51: (1,3,1,1) 132: (5,3) 252: (1,1,1,1,1,3)
(End)
Numbers whose binary representation has the property that every run of consecutive 0's has even length. - Harry Richman, Jan 31 2024
|