A026317 -id:A026317 - cp's OEIS frontend

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

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 *)

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 *)

A026319 a(n) is the n-th nonnegative integer k satisfying |sin(k)| < |cos(k)| < |sin(k+1)|.

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, 186, 189, 192, 208, 211, 214, 230, 233, 236, 252, 255, 258, 274, 277, 280, 296, 299, 302

Offset: 1

Clark Kimberling

nonn

Cf. A026317, A246303, A327138, A327139.

Select[Range[0,310],Abs[Sin[#]]Stefano Spezia, Feb 04 2025 *)

cp's OEIS Frontend

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

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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).

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

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A026319 a(n) is the n-th nonnegative integer k satisfying |sin(k)| < |cos(k)| < |sin(k+1)|.

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