A233694 Position of n in the sequence (or tree) S generated in order by these rules: 0 is in S; if x is in S then x + 1 is in S; if nonzero x is in S then 1/x is in S; if x is in S, then i*x is in S; where duplicates are deleted as they occur.

1, 2, 3, 5, 11, 23, 49, 102, 212, 443, 926, 1939, 4064, 8509, 17816, 37303, 78105, 163544, 342454, 717076, 1501502, 3144024, 6583334, 13784969

Offset: 0

Author

Clark Kimberling, Dec 19 2013

Keywords

• nonn
• more

Comments

It can be proved using the division algorithm for Gaussian integers that S is the set of Gaussian rational numbers: (b + c*i)/d, where b,c,d are integers and d is not 0.

The differences of this sequence give the number of elements in each level of the tree. This means that d(n) = a(n) - a(n-1) is at least 1, and is bounded by 3*d(n-1), since there are three times as many elements in each level, before we exclude repetitions. - Jack W Grahl, Aug 10 2018

Examples

			The first 16 numbers generated are as follows: 0, 1, 2, i, 3, 1/2, 2 i, 1 + i, -i, -1, 4, 1/3, 3 i, 3/2, i/2, 1 + 2 i. The positions of the nonnegative integers are 1, 2, 3, 5, 11.
		

Crossrefs

Cf. A233695, A233696, A232559, A226130, A232723, A226080.

Programs

Mathematica

Off[Power::infy]; x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, 1/x, I*x} /. ComplexInfinity -> 0]]], {18}]; On[Power::infy]; t1 = Flatten[Position[x, _?(IntegerQ[#] && NonNegative[#] &)]]    (* A233694 *)
t2 = Flatten[Position[x, _?(IntegerQ[#] && Negative[#] &)]]  (* A233695 *)
t = Union[t1, t2]  (* A233696 *)
(* Peter J. C. Moses, Dec 21 2013 *)

Extensions

More terms from Jack W Grahl, Aug 10 2018