A327137 -id:A327137 - cp's OEIS frontend

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

Clark Kimberling

nonn

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

Cf. A026309, A246303, A327138.

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

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

A327136 Numbers k such that sin(2k) > sin(2k+2) < sin(2k+4).

Original entry on oeis.org

1, 4, 8, 11, 14, 17, 20, 23, 26, 30, 33, 36, 39, 42, 45, 48, 52, 55, 58, 61, 64, 67, 70, 74, 77, 80, 83, 86, 89, 92, 96, 99, 102, 105, 108, 111, 114, 118, 121, 124, 127, 130, 133, 136, 140, 143, 146, 149, 152, 155, 158, 162, 165, 168, 171, 174, 177, 180, 184

Offset: 1

Clark Kimberling, Aug 23 2019

The sequences A026317, A327136, A327137 partition the nonnegative integers.

Conjecture: 1.285 < n*Pi - a(n) < 1.286 for n >= 1.

			(sin 2, sin 4, ...) = (0.9, -0.7, -0.2, 0.9, -0.5, ...) approximately, so that the differences, in sign, are - + + -  + + - - + - - + ..., with "+" in places 2,3,5,6,... (A026317), "- +" starting in places 1,4,8,11,... (A327136), and "- - +" starting in places 7,10,13,16,... (A327137).

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A026309, A246303, A026317.

z = 500; f[x_] := f[x] = Sin[2 x]; t = Range[1, z];
Select[t, f[#] < f[# + 1] &]    (* A026317 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A327136 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]   (* A327137 *)

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

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