1,1

Fanfolding the triangles defined by the spiral of Theodorus, each successive hypotenuse converges to a limit which divides the angle pi/4 of the first triangle. The ratio of the limit angle to pi/4 is .5721501037... = -sum[((-1)^n)(arctan(1/sqrt(n))] / (pi/4) The index n starts at 1 for the first triangle (pi/4), and counts triangles.

I have developed an algorithm to compute the alternating sum, which converges quite slowly. I have computed a value to the precision limit of 64 bit floating point, which is about 14 decimal places. The execution time is about one second. It is unclear whether the algorithm readily adapts to arbitrary precision calculation software in a usable way. There are related sequences: the actual convergent angle in degrees, or radians, etc. as well as a whole family of similarly defined convergents. Other than general treatments of the spiral of Theodorus, no references are known.

.5721501037... = - Sum_{ n >= 1} [((-1)^n)(arctan(1/sqrt(n))] / (pi/4) or = -argument(product[(sqrt(n)+((-1)^n)i)/magnitude(sqrt(n)+((-1)^n)i)]) / (pi/4)

Sequence in context: A108763 A061415 A087455 this_sequence A021640 A099283 A019987

Adjacent sequences: A117756 A117757 A117758 this_sequence A117760 A117761 A117762

cons,hard,more,nonn

Peter Hammer Apr 14 2006