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Template:Sequence of the Day for January 4

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Intended for: January 4, 2012

Timetable

  • First draft entered by Alonso del Arte on November 3, 2011
  • Draft reviewed by Daniel Forgues on November 8, 2011
  • Draft to be approved by December 4, 2011
Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.



A198683: Number of distinct values taken by
i  ^  i  ^    ^  i
(with
n
i
 ’s and
n  −  2
pairs of parentheses inserted in all possible nontrivial ways) where
i
is the imaginary unit and
  ^  
denotes the principal value of the power function.
{ 1, 1, 2, 3, 7, 15, 34, 77, ... }
In the case of say,
n = 4
, we’re considering the expressions
i  ^  (i  ^  (i  ^  i)), i  ^  ((i  ^  i)  ^  i)  =  ((i  ^  i)  ^  i)  ^  i, (i  ^  i)  ^  (i  ^  i)  =  (i  ^  (i  ^  i))  ^  i,

viz.

i (i (ii  )  ), i ((ii  )i  )  =  ((ii  )i  )i, (ii  ) (ii  )  =  (i (ii  ) )i,
and whether or not these expressions result in distinct values or not. As it turns out, the second and third expressions give the same value
i  − i = exp (π / 2)
(approximately 4.81047738...), while the fourth and fifth expressions both result in
(exp ( −  π / 2))  ^  (exp ( −  π / 2))
, (approximately 0.721418...). (By the way, only the first of these expressions results in a number with a nonzero imaginary part.)