A246303 -id:A246303 - cp's OEIS frontend

A026311 n-th nonnegative integer k satisfying cos(k) > cos(k+1).

0, 1, 2, 6, 7, 8, 13, 14, 15, 19, 20, 21, 25, 26, 27, 31, 32, 33, 34, 38, 39, 40, 44, 45, 46, 50, 51, 52, 57, 58, 59, 63, 64, 65, 69, 70, 71, 75, 76, 77, 78, 82, 83, 84, 88, 89, 90, 94, 95, 96, 101, 102, 103, 107, 108, 109, 113, 114, 115

Offset: 1

Clark Kimberling

The sequences A026311, A246300, A246301, A246302 partition the nonnegative integers.

Clark Kimberling, Table of n, a(n) for n = 1..4000

Cf. A246293, A246300, A246301, A246302, A246303 (complement of A026311).

z = 520; f[x_] := f[x] = Cos[x]; t = Range[0, z];
Select[t, f[#] > f[# + 1] &]  (* A026311 *)
Select[t, f[#] < f[# + 1] > f[# + 2] &]  (* A246300 *)
Select[t, f[#] < f[# + 1] < f[# + 2] > f[# + 3] &]  (* A246301 *)
Select[t, f[#] < f[# + 1] < f[# + 2] < f[# + 3] > f[# + 4] &] (* A246302 *)
Join[{0},Position[Partition[Cos[Range[120]],2,1],?(#[[1]]>#[[2]]&),1,Heads -> False]//Flatten] (* _Harvey P. Dale, Aug 09 2021 *)

Comment, Mathematica, Crossrefs by Clark Kimberling, Aug 22 2014

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

Original entry on oeis.org

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 *)

A327138 Numbers k such that cos(2k) < cos(2k+2).

Original entry on oeis.org

2, 5, 8, 11, 12, 14, 15, 17, 18, 20, 21, 24, 27, 30, 33, 34, 36, 37, 39, 40, 42, 43, 46, 49, 52, 55, 56, 58, 59, 61, 62, 64, 65, 68, 71, 74, 77, 78, 80, 81, 83, 84, 86, 87, 90, 93, 96, 99, 100, 102, 103, 105, 106, 108, 109, 112, 115, 118, 121, 122, 124, 125

Offset: 1

Clark Kimberling, Aug 23 2019

The sequences A327138, A327139, A327140 partition the positive integers.

Conjecture: 2.07 < n*Pi - a(n) < 3.08 for n >= 1.

			(cos 2, cos 4, ...) = (-0.4, -0.6, 0.9, -0.1, -0.8, ...) approximately, so that the differences, in sign, are - + - - + - - + - - + +, with "+" in places 2,5,8,11,12,... (A327138), "- +" starting in places 1,4,7,10,13,... (A327139), and "- - +" starting in places 3,6,9,22,25,... (A327140).

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

Cf. A026309, A246303, A026317, A327139.

z = 500; f[x_] := f[x] = Cos[2 x]; t = Range[1, z];
Select[t, f[#] < f[# + 1] &]    (* A327138 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A327139 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]   (* A327140 *)
Position[Partition[Cos[2Range[200]],2,1],?(#[[1]]<#[[2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Oct 21 2024 *)

A246304 Numbers k such that cos(k) > cos(k+1) < cos(k+2).

Original entry on oeis.org

2, 8, 15, 21, 27, 34, 40, 46, 52, 59, 65, 71, 78, 84, 90, 96, 103, 109, 115, 122, 128, 134, 140, 147, 153, 159, 166, 172, 178, 184, 191, 197, 203, 209, 216, 222, 228, 235, 241, 247, 253, 260, 266, 272, 279, 285, 291, 297, 304, 310, 316, 323, 329, 335, 341

Offset: 1

Clark Kimberling, Aug 22 2014

The sequences A246303, A246304, A246305, A246306 partition the nonnegative integers.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A026303, A246305, A246306, A026311 (complement of A246303).

z = 500; f[x_] := f[x] = Cos[x]; t = Range[0, z];
Select[t, f[#] < f[# + 1] &]  (* A246303 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A246304 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]  (* A246305 *)
Select[t, f[#] > f[# + 1] > f[# + 2] > f[# + 3] < f[# + 4] &] (* A246306 *)
Flatten[Position[Partition[Cos[Range[400]],3,1],?(#[[1]]>#[[2]] && #[[2]]< #[[3]]&),{1},Heads->False]] (* _Harvey P. Dale, Apr 23 2015 *)

A246305 Numbers k such that cos(k) > cos(k+1) > cos(k+2) < cos(k+3).

Original entry on oeis.org

1, 7, 14, 20, 26, 33, 39, 45, 51, 58, 64, 70, 77, 83, 89, 95, 102, 108, 114, 121, 127, 133, 139, 146, 152, 158, 165, 171, 177, 183, 190, 196, 202, 208, 215, 221, 227, 234, 240, 246, 252, 259, 265, 271, 278, 284, 290, 296, 303, 309, 315, 322, 328, 334, 340

Offset: 1

Clark Kimberling, Aug 22 2014

The sequences A246303, A246304, A246305, A246306 partition the nonnegative integers.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A026303, A246304, A246306, A026311 (complement of A246303).

z = 500; f[x_] := f[x] = Cos[x]; t = Range[0, z];
Select[t, f[#] < f[# + 1] &]  (* A246303 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A246304 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]  (* A246305 *)
Select[t, f[#] > f[# + 1] > f[# + 2] > f[# + 3] < f[# + 4] &] (* A246306 *)

Definition corrected by Georg Fischer, Apr 01 2024

A246306 Numbers k such that cos(k) > cos(k+1) > cos(k+2) > cos(k+3) < cos(k+4).

Original entry on oeis.org

0, 6, 13, 19, 25, 32, 38, 44, 50, 57, 63, 69, 76, 82, 88, 94, 101, 107, 113, 120, 126, 132, 138, 145, 151, 157, 164, 170, 176, 182, 189, 195, 201, 207, 214, 220, 226, 233, 239, 245, 251, 258, 264, 270, 277, 283, 289, 295, 302, 308, 314, 321, 327, 333, 339

Offset: 1

Clark Kimberling, Aug 22 2014

The sequences A246303, A246304, A246305, A246306 partition the nonnegative integers.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A026303, A246304, A246305, A026311 (complement of A246303).

z = 500; f[x_] := f[x] = Cos[x]; t = Range[0, z];
Select[t, f[#] < f[# + 1] &]  (* A246303 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A246304 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]  (* A246305 *)
Select[t, f[#] > f[# + 1] > f[# + 2] > f[# + 3] < f[# + 4] &] (* A246306 *)

Definition corrected by Georg Fischer, Apr 01 2024

A327139 Numbers k such that cos(2k) > cos(2k+2) < cos(2k+4).

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 19, 23, 26, 29, 32, 35, 38, 41, 45, 48, 51, 54, 57, 60, 63, 67, 70, 73, 76, 79, 82, 85, 89, 92, 95, 98, 101, 104, 107, 111, 114, 117, 120, 123, 126, 129, 133, 136, 139, 142, 145, 148, 151, 155, 158, 161, 164, 167, 170, 173, 176, 180, 183

Offset: 1

Clark Kimberling, Aug 23 2019

The sequences A327138, A327139, A327140 partition the positive integers.

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

Cf. A026309, A246303, A026317, A327138.

z = 500; f[x_] := f[x] = Cos[2 x]; t = Range[1, z];
Select[t, f[#] < f[# + 1] &]    (* A327138 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A327139 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]   (* A327140 *)

(cos 2, cos 4, ...) = (-0.4, -0.6, 0.9, -0.1, -0.8, ...) approximately, so that the differences, in sign, are - + - - + - - + - - + +, with "+" in places 2,5,8,11,12,... (A327138), "- +" starting in places 1,4,7,10,13,... (A327139), and "- - +" starting in places 3,6,9,22,25,... (A327140).

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 *)

A327137 Numbers k such that sin(2k) > sin(2k+2) > sin(2k+4) < sin(2k+6).

Original entry on oeis.org

7, 10, 13, 16, 29, 32, 35, 38, 51, 54, 57, 60, 73, 76, 79, 82, 95, 98, 101, 104, 117, 120, 123, 126, 139, 142, 145, 148, 161, 164, 167, 170, 183, 186, 189, 192, 205, 208, 211, 214, 227, 230, 233, 236, 249, 252, 255, 258, 271, 274, 277, 280, 293, 296, 299

Offset: 1

Clark Kimberling, Aug 23 2019

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

			(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 *)

A327140 Numbers k such that cos(2k) > cos(2k+2) > cos(2k+4) < cos(2k+6).

Original entry on oeis.org

3, 6, 9, 22, 25, 28, 31, 44, 47, 50, 53, 66, 69, 72, 75, 88, 91, 94, 97, 110, 113, 116, 119, 132, 135, 138, 141, 154, 157, 160, 163, 179, 182, 185, 188, 201, 204, 207, 210, 223, 226, 229, 232, 245, 248, 251, 254, 267, 270, 273, 276, 289, 292, 295, 298, 311

Offset: 1

Clark Kimberling, Aug 23 2019

The sequences A327138, A327139, A327140 partition the positive integers.

			(cos 2, cos 4, ...) = (-0.4, -0.6, 0.9, -0.1, -0.8, ...) approximately, so that the differences, in sign, are - + - - + - - + - - + +, with "+" in places 2,5,8,11,12, ... (A327138), "- +" starting in places 1,4,7,10,13,... (A327139), and "- - +" starting in places 3,6,9,22,25,... (A327140).

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

Cf. A026309, A246303, A026317, A327138.

z = 500; f[x_] := f[x] = Cos[2 x]; t = Range[1, z];
Select[t, f[#] < f[# + 1] &]    (* A327138 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A327139 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]   (* A327140 *)

cp's OEIS Frontend

A026311 n-th nonnegative integer k satisfying cos(k) > cos(k+1).

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

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A327138 Numbers k such that cos(2k) < cos(2k+2).

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A246304 Numbers k such that cos(k) > cos(k+1) < cos(k+2).

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A246305 Numbers k such that cos(k) > cos(k+1) > cos(k+2) < cos(k+3).

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A246306 Numbers k such that cos(k) > cos(k+1) > cos(k+2) > cos(k+3) < cos(k+4).

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A327139 Numbers k such that cos(2k) > cos(2k+2) < cos(2k+4).

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

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A327137 Numbers k such that sin(2k) > sin(2k+2) > sin(2k+4) < sin(2k+6).