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.

1, 2, 3, 5, 9, 16, 27, 42, 61, 84, 111, 142, 177, 216, 259, 306, 357, 412, 471, 534, 601, 672, 747, 826, 909, 996, 1087, 1182, 1281, 1384, 1491, 1602, 1717, 1836, 1959, 2086, 2217, 2352, 2491, 2634, 2781, 2932, 3087, 3246, 3409, 3576, 3747, 3922, 4101, 4284

Offset

1,2

Comments

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.

Links

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

Formula

a(n+4) = 2*n^2 + n + 6 for n >= 1 (conjectured).

G.f.: (-1 + x - x^3 - x^4 - x^5 - x^6)/(x -1)^3 (conjectured).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 8 (conjectured).

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A232559, A232867, A232868.

Sequence in context: A124642 A370641 A269153 * A011826 A119968 A291311

Adjacent sequences: A232863 A232864 A232865 * A232867 A232868 A232869

Keyword

nonn,easy

Author

Clark Kimberling, Dec 01 2013

Status

approved