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%I #23 Sep 08 2022 08:44:49
%S 0,2,3,5,6,9,12,15,18,19,21,22,24,25,27,28,31,34,37,40,41,43,44,46,47,
%T 49,50,53,56,59,62,63,65,66,68,69,71,72,75,78,81,84,85,87,88,90,91,93,
%U 94,97,100,103,106,107,109,110,112,113,115
%N Nonnegative integers k such that |cos(k)| > |sin(k+1)|.
%C The sequences A026317, A327136 and A327137 partition the nonnegative integers. - _Clark Kimberling_, Aug 23 2019
%C Requirement can be rewritten cos^2(k) > sin^2(k+1) => cos^2(k) > 1-cos^2(k+1) => cos^2(k+1) > 1-cos^2(k) => |cos(k+1)| > |sin(k)|. - _R. J. Mathar_, Sep 03 2019
%C These are also the numbers k such that sin(2k) < sin(2k+2).
%C Proof (Jean-Paul Allouche, Nov 14 2019):
%C cos^2(n) > sin^2(n+1) ;
%C Formulas for squares Abramowitz-Stegun 4.3.31 and 4.3.32:
%C 1/2 + cos(2n)/2 > 1/2 - cos(2n+2) ;
%C cos(2n+2) + cos(2n) > 0 ;
%C Formulas for sums Abramowitz-Stegun 4.3.16 and 4.3.17:
%C cos(2n)*cos(2) - sin(2n)*sin(2) + cos(2n) > 0 ;
%C (1+cos(2))*cos(2n) > sin(2n)*sin 2;
%C Multiply both sides by 1-cos(2) which is >0:
%C (1-cos^2(2))*cos(2n) > (1-cos(2))*sin(2)*sin(2n) ;
%C sin^2(2)*cos(2n) > (1-cos(2))*sin(2)*sin(2n) ;
%C sin(2)*cos(2n) > (1-cos(2))*sin(2n) ;
%C (1-cos(2))*sin(2n) < cos(2n)*sin 2 ;
%C sin(2n) - sin(2n)*cos(2) < cos(2n)*sin(2);
%C sin(2n) < sin(2n)*cos(2)+cos(2n)*sin(2);
%C And backward application of Abramowitz-Stegun 4.3.16
%C sin(2n) < sin(2n+2) q.e.d.
%C Also nonnegative integers k such that cos(2k+1) > 0. Note that sin(2k+2) - sin(2k) = 2*cos(2k+1)*sin(1). - _Jianing Song_, Nov 16 2019
%t Select[Range[0,120],Abs[Cos[#]]>Abs[Sin[#+1]]&] (* _Harvey P. Dale_, Mar 04 2013 *)
%o (Magma) [k:k in [0..120]|Abs(Cos(k)) gt Abs(Sin(k+1))]; // _Marius A. Burtea_, Nov 14 2019
%Y Cf. A026309, A246303, A327138.
%K nonn
%O 1,2
%A _Clark Kimberling_
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