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A232867
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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.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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