A244658 a(n) = x/(x1-floor(x1)) where x = sqrt(n) - floor(sqrt(n)), x1 = 1/x, a(n) = -1 if division by zero, a(n) = 0 for nonintegers.

-1, 1, 2, -1, 1, 2, 0, 4, -1, 1, 2, 3, 0, 0, 6, -1, 1, 2, 0, 4, 0, 0, 0, 8, -1, 1, 2, 0, 0, 5, 0, 0, 0, 0, 10, -1, 1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12, -1, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 14, -1, 1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 16, -1, 1, 2, 3, 0, 0, 6, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 18, -1, 1, 2, 0, 4, 5

Offset

1,3

Comments

Inspired by the square root of three poem in "Harold and Kumar" the movie. This operation can be done in iterations for which some n seem to give all integer results after iteration k >= 2. Some of them have periodic properties similar to that of the continued fraction method (but not exactly the same). When a(n) is arranged as a table read by rows, the row sums would be A062731. See illustrations in links.

Links

Table of n, a(n) for n=1..105.

Kival Ngaokrajang, Illustration for n = 1..20 and iteration k = 1..10

  a(n) arrange as table for n = 1..168

answers.yahoo, What are the words to the math poem in "Harold and Kumar"?

Eric Weisstein's World of Mathematics, Periodic Continued Fraction

Example

For n = 3, x = sqrt(3) - floor(sqrt(3)) = 0.732050807..., x1 = 1/x = 1/0.732050807... = 1.366025403..., x1 - floor(x1) =  0.366025403..., a(3) = 0.732050807.../0.366025402... = 2.

Prog

(Small Basic)
For n=1 To 500
  x=math.Power(n, .5)
  y=x-math.Floor(x)
  If y<>0 Then
    x1=1/y
    x2=1/(x1-math.Floor(x1))
    a1=x2/x1
    a2=a1-math.floor(a1)
    If a2 > 0.999999 or a2 < 0.0000001 then
      a=math.Round(a1)
      TextWindow.Write(a+", ")
    Else
      TextWindow.Write(0+", ")'noninteger
    Endif
  Else
    TextWindow.Write(-1+", ")'zero division, Square number
  EndIf
EndFor

Crossrefs

Cf. A062731.

Sequence in context: A088226 A344309 A358338 * A117586 A307988 A268917

Adjacent sequences: A244655 A244656 A244657 * A244659 A244660 A244661

Keyword

sign

Author

Kival Ngaokrajang, Jul 04 2014

Status

approved