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Positions of the integers in the sequence (or tree) of complex numbers generated by these rules: 0 is in S, and if x is in S, then x + 1 and i*x are in S, where duplicates are deleted as they occur.
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%I #7 Feb 20 2018 22:50:20

%S 1,2,3,5,8,9,12,16,19,27,30,42,45,61,64,84,87,111,114,142,145,177,180,

%T 216,219,259,262,306,309,357,360,412,415,471,474,534,537,601,604,672,

%U 675,747,750,826,829,909,912,996,999,1087,1090,1182,1185,1281,1284

%N Positions of the integers in the sequence (or tree) of complex numbers generated by these rules: 0 is in S, and if x is in S, then x + 1 and i*x are in S, where duplicates are deleted as they occur.

%C Let S be the sequence (or tree) of complex numbers defined by these rules: 0 is in S, and if x is in S, then x + 1, and i*x are in S. Deleting duplicates as they occur, the generations of S are given by g(1) = (0), g(2) = (1), g(3) = (2,i), g(4) = (3, 2i, 1+i, -1), ... Concatenating these gives 0, 1, 2, i, 3, 2*i, 1 + i, -1, 4, 3*i, 1 + 2*i, -2, 2 + i, -1 + i, -i, 5, ... A232868 is the (ordered) union of two linearly recurrent sequences: A232866 and A232867.

%F From _Chai Wah Wu_, Feb 20 2018: (Start)

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 10 (conjectured).

%F G.f.: x*(-x^9 - 4*x^7 + 2*x^6 + 2*x^5 - 2*x^4 + x^2 - x - 1)/((x - 1)^3*(x + 1)^2) (conjectured). (End)

%e Each x begets x + 1, and i*x, but if either these has already occurred it is deleted. Thus, 0 begets (1); then 1 begets (2,i,); then 2 begets 3 and 2*i, and i begets 1 + i and -1, so that g(4) = (3, 2*i, 1 + i, -1), etc.

%t x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, I*x}]]], {40}]; x;

%t t1 = Flatten[Table[Position[x, n], {n, 0, 30}]] (* A232866 *)

%t t2 = Flatten[Table[Position[x, -n], {n, 1, 30}]] (* A232867 *)

%t Union[t1, t2] (* A232868 *)

%Y Cf. A232559, A232866, A232867.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Dec 01 2013