A232866 -id:A232866 - cp's OEIS frontend

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.

1, 2, 3, 5, 8, 9, 12, 16, 19, 27, 30, 42, 45, 61, 64, 84, 87, 111, 114, 142, 145, 177, 180, 216, 219, 259, 262, 306, 309, 357, 360, 412, 415, 471, 474, 534, 537, 601, 604, 672, 675, 747, 750, 826, 829, 909, 912, 996, 999, 1087, 1090, 1182, 1185, 1281, 1284

Offset: 1

Clark Kimberling, Dec 01 2013

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.

			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.

Cf. A232559, A232866, A232867.

x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, I*x}]]], {40}]; x;
t1 = Flatten[Table[Position[x, n], {n, 0, 30}]]   (* A232866 *)
t2 = Flatten[Table[Position[x, -n], {n, 1, 30}]]  (* A232867 *)
Union[t1, t2]  (* A232868 *)

From Chai Wah Wu, Feb 20 2018: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 10 (conjectured).
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)

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.

Original entry on oeis.org

8, 12, 19, 30, 45, 64, 87, 114, 145, 180, 219, 262, 309, 360, 415, 474, 537, 604, 675, 750, 829, 912, 999, 1090, 1185, 1284, 1387, 1494, 1605, 1720, 1839, 1962, 2089, 2220, 2355, 2494, 2637, 2784, 2935, 3090, 3249, 3412, 3579, 3750, 3925, 4104, 4287, 4474

Offset: 1

Clark Kimberling, Dec 01 2013

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.

			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.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A232559, A232866, A232868.

x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, I*x}]]], {40}]; x;
t1 = Flatten[Table[Position[x, n], {n, 0, 30}]]    (* A232866 *)
t2 = Flatten[Table[Position[x, -n], {n, 1, 30}]]  (* A232867 *)
Union[t1, t2]  (* A232868 *)

a(n+1) = 2*n^2 + n + 9 for n >= 1 (conjectured).
G.f.:  (-8 + 12 x - 7 x^2 - x^3)/(x -1)^3 (conjectured).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 5 (conjectured).

cp's OEIS Frontend

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.

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