cp's OEIS Frontend

A232867 Positions of the negative 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 A232867 #4 Dec 03 2013 13:14:18
%S A232867 8,12,19,30,45,64,87,114,145,180,219,262,309,360,415,474,537,604,675,
%T A232867 750,829,912,999,1090,1185,1284,1387,1494,1605,1720,1839,1962,2089,
%U A232867 2220,2355,2494,2637,2784,2935,3090,3249,3412,3579,3750,3925,4104,4287,4474
%N A232867 Positions of the negative 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 A232867 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 A232867 Clark Kimberling, <a href="/A232867/b232867.txt">Table of n, a(n) for n = 1..1000</a>
%F A232867 a(n+1) = 2*n^2 + n + 9 for n >= 1 (conjectured).
%F A232867 G.f.:  (-8 + 12 x - 7 x^2 - x^3)/(x -1)^3 (conjectured).
%F A232867 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 5 (conjectured).
%e A232867 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.; a(1) = 8 because -1 occurs in the 8th position of S.
%t A232867 x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, I*x}]]], {40}]; x;
%t A232867 t1 = Flatten[Table[Position[x, n], {n, 0, 30}]]    (* A232866 *)
%t A232867 t2 = Flatten[Table[Position[x, -n], {n, 1, 30}]]  (* A232867 *)
%t A232867 Union[t1, t2]  (* A232868 *)
%Y A232867 Cf. A232559, A232866, A232868.
%K A232867 nonn,easy
%O A232867 1,1
%A A232867 _Clark Kimberling_, Dec 01 2013