

A233695


a(n) gives the 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.


4



10, 18, 30, 56, 109, 219, 450, 933, 1946, 4071, 8516, 17823, 37310, 78112, 163551, 342461, 717083, 1501509, 3144031, 6583341, 13784976
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OFFSET

1,1


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.
Empirically, it appears that a(n) = A233694(n+2) + 7 for n > 2. It seems clear that positive integers appear for the first time at the start of a new level of the tree. If this is always the case, then the row starting with n will be followed by a row starting n+1, 1/n, ni, followed by a row starting n+2, 1/(n+1), (n+1)i, 1+1/n, n+1, i/(n+1), 1+ni, i/n, n. It may be possible to show that of these 9 values, only n+1 has ever appeared before. If so, then n will always appear exactly 7 places after n + 2 in the sequence.  Jack W Grahl, Aug 10 2018


LINKS

Table of n, a(n) for n=1..21.
Jack W Grahl, Haskell code to generate this sequence


EXAMPLE

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. 1 appears in the 10th place, so a(1) = 10.


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 *)


CROSSREFS

Cf. A233694, A233696, A232559, A226130, A232723, A226080.
Sequence in context: A167607 A162828 A190038 * A014006 A090995 A153360
Adjacent sequences: A233692 A233693 A233694 * A233696 A233697 A233698


KEYWORD

nonn,more


AUTHOR

Clark Kimberling, Dec 19 2013


EXTENSIONS

More terms by Jack W Grahl, Aug 10 2018


STATUS

approved



