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A232868 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.

%I A232868 #7 Feb 20 2018 22:50:20
%S A232868 1,2,3,5,8,9,12,16,19,27,30,42,45,61,64,84,87,111,114,142,145,177,180,
%T A232868 216,219,259,262,306,309,357,360,412,415,471,474,534,537,601,604,672,
%U A232868 675,747,750,826,829,909,912,996,999,1087,1090,1182,1185,1281,1284
%N A232868 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 A232868 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 A232868 From _Chai Wah Wu_, Feb 20 2018: (Start)
%F A232868 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 10 (conjectured).
%F A232868 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 A232868 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 A232868 x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, I*x}]]], {40}]; x;
%t A232868 t1 = Flatten[Table[Position[x, n], {n, 0, 30}]]   (* A232866 *)
%t A232868 t2 = Flatten[Table[Position[x, -n], {n, 1, 30}]]  (* A232867 *)
%t A232868 Union[t1, t2]  (* A232868 *)
%Y A232868 Cf. A232559, A232866, A232867.
%K A232868 nonn,easy
%O A232868 1,2
%A A232868 _Clark Kimberling_, Dec 01 2013