A026309 -id:A026309 - cp's OEIS frontend

A246293 Numbers k such that sin(k) > sin(k+1).

2, 3, 4, 8, 9, 10, 14, 15, 16, 20, 21, 22, 23, 27, 28, 29, 33, 34, 35, 39, 40, 41, 46, 47, 48, 52, 53, 54, 58, 59, 60, 64, 65, 66, 67, 71, 72, 73, 77, 78, 79, 83, 84, 85, 90, 91, 92, 96, 97, 98, 102, 103, 104, 108, 109, 110, 111, 115, 116, 117, 121, 122, 123

Offset: 1

Clark Kimberling, Aug 21 2014

The sequences A246293, A246294, A246295, A246296 partition the nonnegative integers.

Numbers like 42, 86, 130, 199, 243, 287,.. are in none of these 4 sequences. - R. J. Mathar, May 18 2020

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

Cf. A246294, A246295, A246296, A026309 (complement of A246293).

z = 500; f[x_] := f[x] = Sin[x]; t = Range[0, z];
Select[t, f[#] > f[# + 1] &]  (* A246293 *)
Select[t, f[#] < f[# + 1] > f[# + 2] &]  (* A246294 *)
Select[t, f[#] < f[# + 1] < f[# + 2] > f[# + 3] &]  (* A246295 *)
Select[t, f[#] < f[# + 1] < f[# + 2] < f[# + 3] > f[# + 4] &] (* A246296 *)
Position[Partition[Sin[Range[130]],2,1],?(#[[1]]>#[[2]]&),{1},Heads-> False]//Flatten (* _Harvey P. Dale, Sep 18 2016 *)

is(n)=sin(n)>sin(n+1) \\ Charles R Greathouse IV, Aug 03 2017

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 *)

A246294 Numbers k such that sin(k) < sin(k+1) > sin(k+2).

Original entry on oeis.org

1, 7, 13, 19, 26, 32, 38, 45, 51, 57, 63, 70, 76, 82, 89, 95, 101, 107, 114, 120, 126, 133, 139, 145, 151, 158, 164, 170, 176, 183, 189, 195, 202, 208, 214, 220, 227, 233, 239, 246, 252, 258, 264, 271, 277, 283, 290, 296, 302, 308, 315, 321, 327, 334, 340

Offset: 1

Clark Kimberling, Aug 21 2014

The sequences A246293, A246294, A246295, A246296 partition the nonnegative integers.

Numbers like 42, 86, 130, 199, 243, 287,.. are in none of these 4 sequences. - R. J. Mathar, May 18 2020

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

Cf. A246293, A246295, A246296, A026309 (complement of A246293).

z = 500; f[x_] := f[x] = Sin[x]; t = Range[0, z];
Select[t, f[#] > f[# + 1] &]  (* A246293 *)
Select[t, f[#] < f[# + 1] > f[# + 2] &]  (* A246294 *)
Select[t, f[#] < f[# + 1] < f[# + 2] > f[# + 3] &]  (* A246295 *)
Select[t, f[#] < f[# + 1] < f[# + 2] < f[# + 3] > f[# + 4] &] (* A246296 *)
Position[Partition[Sin[Range[350]],3,1],?(#[[1]]<#[[2]]>#[[3]]&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Aug 03 2017 *)

A246295 Numbers k such that sin(k) < sin(k+1) < sin(k+2) > sin(k+3).

Original entry on oeis.org

0, 6, 12, 18, 25, 31, 37, 44, 50, 56, 62, 69, 75, 81, 88, 94, 100, 106, 113, 119, 125, 132, 138, 144, 150, 157, 163, 169, 175, 182, 188, 194, 201, 207, 213, 219, 226, 232, 238, 245, 251, 257, 263, 270, 276, 282, 289, 295, 301, 307, 314, 320, 326, 333, 339

Offset: 1

Clark Kimberling, Aug 21 2014

The sequences A246293, A246294, A246295, A246296 partition the nonnegative integers.

Numbers like 42, 86, 130, 199, 243, 287,.. are in none of these 4 sequences. - R. J. Mathar, May 18 2020

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

Cf. A246293, A246294, A246296, A026309 (complement of A246293).

z = 500; f[x_] := f[x] = Sin[x]; t = Range[0, z];
Select[t, f[#] > f[# + 1] &]  (* A246293 *)
Select[t, f[#] < f[# + 1] > f[# + 2] &]  (* A246294 *)
Select[t, f[#] < f[# + 1] < f[# + 2] > f[# + 3] &]  (* A246295 *)
Select[t, f[#] < f[# + 1] < f[# + 2] < f[# + 3] > f[# + 4] &] (* A246296 *)

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 *)

A246296 Numbers k such that sin(k) < sin(k+1) < sin(k+2) < sin(k+3) > sin(k+4).

Original entry on oeis.org

5, 11, 17, 24, 30, 36, 43, 49, 55, 61, 68, 74, 80, 87, 93, 99, 105, 112, 118, 124, 131, 137, 143, 149, 156, 162, 168, 174, 181, 187, 193, 200, 206, 212, 218, 225, 231, 237, 244, 250, 256, 262, 269, 275, 281, 288, 294, 300, 306, 313, 319, 325, 332, 338, 344

Offset: 1

Clark Kimberling, Aug 21 2014

The sequences A246293, A246294, A246295, A246296 partition the nonnegative integers.

Numbers like 42, 86, 130, 199, 243, 287,.. are in none of these 4 sequences. - R. J. Mathar, May 18 2020

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

Cf. A246293, A246294, A246295, A026309 (complement of A246293).

z = 500; f[x_] := f[x] = Sin[x]; t = Range[0, z];
Select[t, f[#] > f[# + 1] &]  (* A246293 *)
Select[t, f[#] < f[# + 1] > f[# + 2] &]  (* A246294 *)
Select[t, f[#] < f[# + 1] < f[# + 2] > f[# + 3] &]  (* A246295 *)
Select[t, f[#] < f[# + 1] < f[# + 2] < f[# + 3] > f[# + 4] &] (* A246296 *)

A246297 Numbers k such that sin(k) > sin(k+1) < sin(k+2).

Original entry on oeis.org

4, 10, 16, 23, 29, 35, 41, 48, 54, 60, 67, 73, 79, 85, 92, 98, 104, 111, 117, 123, 129, 136, 142, 148, 155, 161, 167, 173, 180, 186, 192, 198, 205, 211, 217, 224, 230, 236, 242, 249, 255, 261, 268, 274, 280, 286, 293, 299, 305, 312, 318, 324, 330, 337, 343

Offset: 1

Clark Kimberling, Aug 21 2014

The sequences A026309, A246297, A246298, A246299 partition the nonnegative integers.

Numbers like 20, 64, 108, 152,... are in none of these 4 sequences. - R. J. Mathar, May 18 2020

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

Cf. A026309, A246298, A246299, A246293 (complement of A026309).

z = 500; f[x_] := f[x] = Sin[x]; t = Range[0, z];
Select[t, f[#] < f[# + 1] &]  (* A026309 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A246297 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]  (* A246298 *)
Select[t, f[#] > f[# + 1] > f[# + 2] > f[# + 3] < f[# + 4] &] (* A246299 *)

Corrected signs in NAME. - R. J. Mathar, May 18 2020

A246298 Numbers k such that sin(k) > sin(k+1) > sin(k+2) < sin(k+3).

Original entry on oeis.org

3, 9, 15, 22, 28, 34, 40, 47, 53, 59, 66, 72, 78, 84, 91, 97, 103, 110, 116, 122, 128, 135, 141, 147, 154, 160, 166, 172, 179, 185, 191, 197, 204, 210, 216, 223, 229, 235, 241, 248, 254, 260, 267, 273, 279, 285, 292, 298, 304, 311, 317, 323, 329, 336, 342

Offset: 1

Clark Kimberling, Aug 21 2014

The sequences A026309, A246297, A246298, A246299 partition the nonnegative integers.

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

Cf. A246297, A246299, A246293 (complement of A026309).

z = 500; f[x_] := f[x] = Sin[x]; t = Range[0, z];
Select[t, f[#] < f[# + 1] &]  (* A026309 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A246297 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]  (* A246298 *)
Select[t, f[#] > f[# + 1] > f[# + 2] > f[# + 3] < f[# + 4] &] (* A246299 *)
Flatten[Position[Partition[Sin[Range[350]],4,1],?(#[[1]]>#[[2]]>#[[3]]<#[[4]]&),1,Heads->False]] (* _Harvey P. Dale, Aug 03 2017 *)

q(n)=my(s0=sin(n),s1=sin(n+1),s2=sin(n+2),s3=sin(n+3));if( (s0>s1) && (s1>s2) && (s2Joerg Arndt, Aug 03 2017

list(lim)=my(v=List(),u=vector(4,x,sin(x+2))); forstep(k=3,lim-3,4, u[4]=sin(k+3); if(u[1]>u[2]&&u[2]>u[3]&&u[3]u[3]&&u[3]>u[4]&&u[4]u[4]&&u[4]>u[1]&&u[1]u[1]&&u[1]>u[2]&&u[2]sin(k+1)&&sin(k+1)>sin(k+2)&&sin(k+2)Charles R Greathouse IV, Aug 03 2017

from sympy import sin
def ok(n):
    s0, s1, s2, s3 = sin(n), sin(n + 1), sin(n + 2), sin(n + 3)
    return s0>s1 and s1>s2 and s2Indranil Ghosh, Aug 03 2017

Name corrected by Harvey P. Dale, Aug 03 2017

A246299 Numbers k such that sin(k) > sin(k+1) > sin(k+2) > sin(k+3) < sin(k+4).

Original entry on oeis.org

2, 8, 14, 21, 27, 33, 39, 46, 52, 58, 65, 71, 77, 83, 90, 96, 102, 109, 115, 121, 127, 134, 140, 146, 153, 159, 165, 171, 178, 184, 190, 196, 203, 209, 215, 222, 228, 234, 240, 247, 253, 259, 266, 272, 278, 284, 291, 297, 303, 310, 316, 322, 328, 335, 341

Offset: 1

Clark Kimberling, Aug 21 2014

The sequences A026309, A246297, A246298, A246299 partition the nonnegative integers.

Numbers like 20, 64, 108, 152, 177, 221, 265, 309, ... are in none of these 4 sequences. - R. J. Mathar, May 18 2020

R. J. Mathar, Table of n, a(n) for n = 1..987

Cf. A026309, A246297, A246298, A246293 (complement of A026309).

z = 500; f[x_] := f[x] = Sin[x]; t = Range[0, z];
Select[t, f[#] < f[# + 1] &]  (* A026309 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A246297 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]  (* A246298 *)
Select[t, f[#] > f[# + 1] > f[# + 2] > f[# + 3] < f[# + 4] &] (* A246299 *)
Position[Partition[Table[Sin[n],{n,400}],5,1],?(#[[1]]>#[[2]]>#[[3]]>#[[4]]<#[[5]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, May 13 2023 *)

Corrected signs in NAME. - R. J. Mathar, May 18 2020

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).

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A246293 Numbers k such that sin(k) > sin(k+1).

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

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A246294 Numbers k such that sin(k) < sin(k+1) > sin(k+2).

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A246295 Numbers k such that sin(k) < sin(k+1) < sin(k+2) > sin(k+3).

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

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A246296 Numbers k such that sin(k) < sin(k+1) < sin(k+2) < sin(k+3) > sin(k+4).

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A246297 Numbers k such that sin(k) > sin(k+1) < sin(k+2).

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A246298 Numbers k such that sin(k) > sin(k+1) > sin(k+2) < sin(k+3).

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