cp's OEIS Frontend

A046956 Numbers k where tan(k) decreases monotonically to 0 (or cot(k) increases).

%I A046956 #24 Aug 17 2020 02:48:24
%S A046956 1,4,7,10,13,16,19,22,355,104348,312689,1146408,5419351,85563208,
%T A046956 165707065,411557987,1480524883,2549491779,8717442233,14885392687,
%U A046956 35938735828,56992078969,78045422110,99098765251,120152108392
%N A046956 Numbers k where tan(k) decreases monotonically to 0 (or cot(k) increases).
%C A046956 From _Jon E. Schoenfield_, Aug 10 2006: (Start)
%C A046956 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.
%C A046956 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.
%C A046956 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:
%C A046956 [3;i], i=1..7 (6 semiconvergents and 1 convergent)
%C A046956 [3;7,15,1]
%C A046956 [3;7,15,1,292,1]
%C A046956 [3;7,15,1,292,1,1,1]
%C A046956 [3;7,15,1,292,1,1,1,2,1]
%C A046956 [3;7,15,1,292,1,1,1,2,1,3,1]
%C A046956 [3;7,15,1,292,1,1,1,2,1,3,1,14,i], i=1..2
%C A046956 [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1]
%C A046956 [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,i], i=1..2
%C A046956 [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,i], i=1..2
%C A046956 [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)
%C A046956 See also A002485 which has a similar property (numerators of convergents to pi = numbers for which |tan a(n)| decreases to zero). - _M. F. Hasler_, Apr 01 2013
%e A046956 a(1) is the numerator of [3;1] = 3 + 1/1 = 4/1
%e A046956 a(2) is the numerator of [3;2] = 3 + 1/2 = 7/2
%e A046956 ...
%e A046956 a(7) is the numerator of [3;7] = 3 + 1/7 = 22/7
%e A046956 a(8) is the numerator of [3;7,15,1] = 3 + 1/(7 + 1/(15 + 1/1)) = 355/113
%e A046956 a(9) is the numerator of [3;7,15,1,292,1] = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + 1/1)))) = 104348/33215
%t A046956 s = Tan[1]; Do[t = Tan[n]; If[t > 0 && t <= s, Print[n]; s = t], {n, 10^9}] (* _Ryan Propper_, Jul 27 2006 *)
%o A046956 (PARI) e=2;for(n=1,1e9, tan(n)>0 && tan(n)<e && !print1(n",") && e=tan(n)) \\ - _M. F. Hasler_, Apr 01 2013
%Y A046956 Cf. A001203, A004112, A002485 (|tan a(n)|->0).
%K A046956 nonn
%O A046956 0,2
%A A046956 _Olivier Gérard_
%E A046956 More terms from _Michel ten Voorde_
%E A046956 2 more terms from _Ryan Propper_, Jul 27 2006
%E A046956 More terms from _Jon E. Schoenfield_, Aug 10 2006
%E A046956 Corrected by _Don Reble_, Nov 20 2006