A232723 Sequence (or tree) generated by these rules: 0 is in S, and if x is in S, then 2*x and 1 - x are in S, and duplicates are deleted as they occur.

0, 1, 2, 4, -1, 8, -3, -2, 16, -7, -6, -4, 3, 32, -15, -14, -12, 7, -8, 5, 6, 64, -31, -30, -28, 15, -24, 13, 14, -16, 9, 10, 12, -5, 128, -63, -62, -60, 31, -56, 29, 30, -48, 25, 26, 28, -13, -32, 17, 18, 20, -9, 24, -11, -10, 256, -127, -126, -124, 63

Offset: 1

Author

Clark Kimberling, Nov 28 2013

Keywords

• sign
• easy

Comments

Let S be the set of numbers defined by these rules: 0 is in S, and if x is in S, then 2*x and 1 - x are in S. Then S is the set of integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (0), g(2) = (1), g(3) = (2), g(4) = (4,-1), g(5) = (8,-3,-2), etc. Concatenating these gives A232723. Every integer occurs exactly once in S. The even integers occupy the positions given by the lower Wythoff sequence, A000201; the odds, by the upper Wythoff sequence, A001950. The positive integers occupy the positions given by A189035, and the positions of the nonpositives, by A189034.

Inverse beginning with 0: 1, 2, 3, 13, 4, 20, 21, 18, 6, 31, 32, 89, 33, 28, 29, 26, 9, 49, 50, 136, 51, 143, 144, 141, 53, 44, ..., . - Robert G. Wilson v, Jun 17 2014

Examples

			Each x begets 2*x and 1 - x, and if either has already occurred it is deleted. Thus, 0 begets 1, which begets 2, which begets (4,-1), etc.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A232559, A226130, A226131, A000201, A189035.

Programs

Mathematica

x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, 2*x, 1 - x}]]], {10}]; x  (* Peter J. C. Moses, Nov 28 2013 *)
Nest[ DeleteDuplicates[ Flatten[ # /. a_Integer -> {2a, 1-a}]]&, {0}, 9] (* Robert G. Wilson v, Jun 17 2014 *)