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

Author

Kival Ngaokrajang, Jul 04 2014

Keywords

• sign

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.

Examples

			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.
		

Links

• 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

Crossrefs

Cf. A062731.