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COMMENT
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Comments from Jon E. Schoenfield (jonscho(AT)hiwaay.net), Aug 10 2006:
(Start) The approach described uses continued fractions containing an even number of terms of which all but the last term are fixed at the values those terms take in the continued fraction for Pi; the final term is initialized at 1 and incremented by 1 each time until it reaches the value taken by that term in the continued fraction for Pi. The semiconvergents and convergents thus obtained are increasingly accurate approximations for Pi, all of which approach Pi from values larger than Pi. Thus the angles whose sizes (in radians) are the numerators of those semiconvergents and convergents approach (from the positive side) integer multiples of Pi, so the tangents of those angles approach zero from positive values.
If we were to use the same approach but with continued fractions having an odd number of terms, i.e. [3] = 3/1; [3;7,i], i=1..15; [3;7,15,1,i], i=1..292; etc.,
then the semiconvergents and convergents obtained would likewise be increasingly accurate approximations for Pi, but they would approach Pi from values smaller than Pi, so the angles whose sizes (in radians) are the numerators of those semiconvergents and convergents would approach (from the negative side) integer multiples of Pi, and thus the tangents of those angles would approach zero from negative values.
Terms after a(0) = 1 are the numerators of the fractions obtained by evaluating all those convergents and semiconvergents of the continued fraction for Pi (A001203) that, as written below, have an even number of partial quotients:
[3;i], i=1..7 (6 semiconvergents and 1 convergent)
[3;7,15,1]
[3;7,15,1,292,1]
[3;7,15,1,292,1,1,1]
[3;7,15,1,292,1,1,1,2,1]
[3;7,15,1,292,1,1,1,2,1,3,1]
[3;7,15,1,292,1,1,1,2,1,3,1,14,i], i=1..2
[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1]
[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,i], i=1..2
[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,i], i=1..2
[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,i], i=1..84, etc. (End)
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