A305369 Lexicographically earliest sequence of distinct positive integers such that for each 1 in the binary expansion of a(n), exactly one of a(n-1) and a(n+1) has a 1 in the same position.

1, 3, 2, 4, 5, 9, 8, 6, 7, 17, 16, 10, 11, 21, 20, 32, 33, 13, 12, 18, 19, 37, 36, 24, 25, 35, 34, 28, 29, 65, 64, 14, 15, 49, 48, 66, 67, 41, 40, 22, 23, 73, 72, 38, 39, 81, 80, 42, 43, 69, 68, 26, 27, 97, 96, 30, 31, 129, 128, 44, 45, 83, 82, 132, 133, 51, 50, 76, 77, 131, 130, 52, 53, 75, 74, 144, 145, 47, 46, 192

Offset: 1

Author

N. J. A. Sloane, Jun 02 2018

Keywords

• nonn
• look

Comments

This is to A280864 as A115510 is to A064413 (EKG) and A252867 is to A098550 (Yellowstone).

Examples

			After a(1) = 1, a(2) is the smallest missing odd number, so a(2) = 3.
a(3) is then the smallest missing number of the form ...1*_2, so a(3) = 10_2 = 2.
After a(15) = 20 = 10100_2, a(16) is the smallest missing number of the form ...0*0**_2, which is 100000_2 = 32.
		

References

• Empirical: a(4k) = 2*Q(2k), a(4k+1) = a(4k)+1, a(4k+2) = 2*Q(2k+1)+1, a(4k+3) = 2*Q(2k+1), where Q (for Quet) is A109812. Since Q has a simpler definition, there is hope for a proof of this connection.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Maple program

Crossrefs

Cf. A280864, A252867, A098550, A115510, A064413, A109812, A352578 (binary weight).

The graphs of A109812, A252867, A305369, A305372 all have roughly the same, mysterious, fractal-like structure. - N. J. A. Sloane, Jun 03 2018