A026317 Nonnegative integers k such that |cos(k)| > |sin(k+1)|.

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, 49, 50, 53, 56, 59, 62, 63, 65, 66, 68, 69, 71, 72, 75, 78, 81, 84, 85, 87, 88, 90, 91, 93, 94, 97, 100, 103, 106, 107, 109, 110, 112, 113, 115

Offset

1,2

Comments

The sequences A026317, A327136 and A327137 partition the nonnegative integers. - Clark Kimberling, Aug 23 2019

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

These are also the numbers k such that sin(2k) < sin(2k+2).

Proof (Jean-Paul Allouche, Nov 14 2019):

cos^2(n) > sin^2(n+1) ;

Formulas for squares Abramowitz-Stegun 4.3.31 and 4.3.32:

1/2 + cos(2n)/2 > 1/2 - cos(2n+2) ;

cos(2n+2) + cos(2n) > 0 ;

Formulas for sums Abramowitz-Stegun 4.3.16 and 4.3.17:

cos(2n)*cos(2) - sin(2n)*sin(2) + cos(2n) > 0 ;

(1+cos(2))*cos(2n) > sin(2n)*sin 2;

Multiply both sides by 1-cos(2) which is >0:

(1-cos^2(2))*cos(2n) > (1-cos(2))*sin(2)*sin(2n) ;

sin^2(2)*cos(2n) > (1-cos(2))*sin(2)*sin(2n) ;

sin(2)*cos(2n) > (1-cos(2))*sin(2n) ;

(1-cos(2))*sin(2n) < cos(2n)*sin 2 ;

sin(2n) - sin(2n)*cos(2) < cos(2n)*sin(2);

sin(2n) < sin(2n)*cos(2)+cos(2n)*sin(2);

And backward application of Abramowitz-Stegun 4.3.16

sin(2n) < sin(2n+2) q.e.d.

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

Links

Table of n, a(n) for n=1..59.

Mathematica

Select[Range[0, 120], Abs[Cos[#]]>Abs[Sin[#+1]]&] (* Harvey P. Dale, Mar 04 2013 *)

Prog

(Magma) [k:k in [0..120]|Abs(Cos(k)) gt Abs(Sin(k+1))]; // Marius A. Burtea, Nov 14 2019

Crossrefs

Cf. A026309, A246303, A327138.

Sequence in context: A295586 A283768 A212864 * A361848 A008768 A067593

Adjacent sequences: A026314 A026315 A026316 * A026318 A026319 A026320

Keyword

nonn

Author

Clark Kimberling

Status

approved