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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.
4
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
1,2
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, ... A232868 is the (ordered) union of two linearly recurrent sequences: A232866 and A232867.
FORMULA
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)
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 01 2013
STATUS
approved