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A232866 Positions of the nonnegative 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 A232866 #5 Dec 03 2013 13:14:02
%S A232866 1,2,3,5,9,16,27,42,61,84,111,142,177,216,259,306,357,412,471,534,601,
%T A232866 672,747,826,909,996,1087,1182,1281,1384,1491,1602,1717,1836,1959,
%U A232866 2086,2217,2352,2491,2634,2781,2932,3087,3246,3409,3576,3747,3922,4101,4284
%N A232866 Positions of the nonnegative 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 A232866 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, ...  It appears that if c and d are integers, than the positions of c*n+d*i, for n>=0, comprise a linear recurrence sequence with signature beginning with 3, -3, 1, following for zero or more 0's.
%H A232866 Clark Kimberling, <a href="/A232866/b232866.txt">Table of n, a(n) for n = 1..1000</a>
%F A232866 a(n+4) = 2*n^2 + n + 6 for n >= 1 (conjectured).
%F A232866 G.f.:  (-1 + x - x^3 - x^4 - x^5 - x^6)/(x -1)^3 (conjectured).
%F A232866 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 8 (conjectured).
%e A232866 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 A232866 x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, I*x}]]], {40}]; x;
%t A232866 t1 = Flatten[Table[Position[x, n], {n, 0, 30}]]   (* A232866 *)
%t A232866 t2 = Flatten[Table[Position[x, -n], {n, 1, 30}]]  (* A232867 *)
%t A232866 Union[t1, t2]  (* A232868 *)
%Y A232866 Cf. A232559, A232867, A232868.
%K A232866 nonn,easy
%O A232866 1,2
%A A232866 _Clark Kimberling_, Dec 01 2013