A246046 -id:A246046 - cp's OEIS frontend

A062389 a(n) = floor( (2n-1)*Pi/2 ).

1, 4, 7, 10, 14, 17, 20, 23, 26, 29, 32, 36, 39, 42, 45, 48, 51, 54, 58, 61, 64, 67, 70, 73, 76, 80, 83, 86, 89, 92, 95, 98, 102, 105, 108, 111, 114, 117, 120, 124, 127, 130, 133, 136, 139, 142, 146, 149, 152, 155, 158, 161, 164, 168, 171, 174, 177, 180, 183, 186

Offset: 1

Jason Earls, Jul 08 2001

In general, the complement of a nonhomogenous Beatty sequence [n*r + h] is given by [n*s + h - h*s], where s = r/(r - 1). As an example, the complement of this sequence is A246046. This sequence gives the positive integers k satisfying tan(k) > tan(k + 1), and A246046 gives those satisfying tan(k) < tan(k + 1). - Clark Kimberling, Aug 24 2014

Excluding a(1), a(n) = positive floored solutions to tan(x) = x. - Derek Orr, May 30 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 223.

Harry J. Smith, Table of n, a(n) for n=1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A246046.

seq(floor((2*n-1)*Pi/2), n=1..1000); # Robert Israel, Jun 01 2015

r = Pi; s = Pi/(Pi - 1); h = -Pi/2; z = 120;
u = Table[Floor[n*r + h], {n, 1, z}] (* A062389 *)
v = Table[Floor[n*s + h - h*s], {n, 1, z}]  (* A246046 *)
(* Clark Kimberling, Aug 24 2014 *)

j=[]; for(n=1,150,j=concat(j,floor(1/2*(2*n-1)*Pi))); j

{ default(realprecision, 50); for (n=1, 1000, write("b062389.txt", n, " ", (2*n - 1)*Pi\2); ) } \\ Harry J. Smith, Aug 06 2009

A246393 Nonnegative integers k satisfying cos(k) >= 0 and cos(k+1) <= 0.

Original entry on oeis.org

1, 7, 14, 20, 26, 32, 39, 45, 51, 58, 64, 70, 76, 83, 89, 95, 102, 108, 114, 120, 127, 133, 139, 146, 152, 158, 164, 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: 0

Clark Kimberling, Aug 24 2014

A246393 and A246394 partition A062389 (the nonhomogeneous Beatty sequence {floor((n-1/2)*Pi)}). Likewise, A246046, the complement of A062389, is partitioned by A246395 and A246396. (See the Mathematica program.)

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A062389, A246394, A246046, A246395, A246396, A246388.

z = 400; f[x_] := Cos[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &]  (* A062389 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &]  (* A246393 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &]  (* A246394 *)
Select[Range[0, z], f[#]*f[# + 1] > 0 &]  (* A246046 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &]  (* A246395 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &]  (* A246396 *)

A246394 Nonnegative integers k satisfying cos(k) <= 0 and cos(k+1) >= 0.

Original entry on oeis.org

4, 10, 17, 23, 29, 36, 42, 48, 54, 61, 67, 73, 80, 86, 92, 98, 105, 111, 117, 124, 130, 136, 142, 149, 155, 161, 168, 174, 180, 186, 193, 199, 205, 212, 218, 224, 230, 237, 243, 249, 256, 262, 268, 274, 281, 287, 293, 300, 306, 312, 318, 325, 331, 337, 344

Offset: 0

Clark Kimberling, Aug 24 2014

A246393 and A246394 partition A062389 (the nonhomogeneous Beatty sequence {floor((n-1/2)*Pi)}). Likewise, A246046, the complement of A062389, is partitioned by A246395 and A246396. (See the Mathematica program.)

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A062389, A246393, A246046, A246395, A246396.

z = 400; f[x_] := Cos[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &]  (* A062389 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &]  (* A246393 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &]  (* A246394 *)
Select[Range[0, z], f[#]*f[# + 1] > 0 &]  (* A246046 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &]  (* A246395 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &]  (* A246396 *)

A246395 Nonnegative integers k satisfying cos(k) >= 0 and cos(k+1) >= 0.

Original entry on oeis.org

0, 5, 6, 11, 12, 13, 18, 19, 24, 25, 30, 31, 37, 38, 43, 44, 49, 50, 55, 56, 57, 62, 63, 68, 69, 74, 75, 81, 82, 87, 88, 93, 94, 99, 100, 101, 106, 107, 112, 113, 118, 119, 125, 126, 131, 132, 137, 138, 143, 144, 145, 150, 151, 156, 157, 162, 163, 169, 170

Offset: 0

Clark Kimberling, Aug 24 2014

A246393 and A246394 partition A062389 (the nonhomogeneous Beatty sequence {floor((n-1/2)*Pi)}). Likewise, A246046, the complement of A062389, is partitioned by A246395 and A246396. (See the Mathematica program.)

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A062389, A246393, A246046, A246394, A246396.

z = 400; f[x_] := Cos[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &]  (* A062389 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &]  (* A246393 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &]  (* A246394 *)
Select[Range[0, z], f[#]*f[# + 1] > 0 &]  (* A246046 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &]  (* A246395 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &]  (* A246396 *)

A246396 Nonnegative integers k satisfying cos(k) <= 0 and cos(k+1) <= 0.

Original entry on oeis.org

2, 3, 8, 9, 15, 16, 21, 22, 27, 28, 33, 34, 35, 40, 41, 46, 47, 52, 53, 59, 60, 65, 66, 71, 72, 77, 78, 79, 84, 85, 90, 91, 96, 97, 103, 104, 109, 110, 115, 116, 121, 122, 123, 128, 129, 134, 135, 140, 141, 147, 148, 153, 154, 159, 160, 165, 166, 167, 172

Offset: 0

Clark Kimberling, Aug 24 2014

A246393 and A246394 partition A062389 (the nonhomogeneous Beatty sequence {floor((n-1/2)*Pi)}). Likewise, A246046, the complement of A062389, is partitioned by A246395 and A246396. (See the Mathematica program.)

Conjecture: every term t has at least one neighbor which is equal to t plus or minus one. - Harvey P. Dale, Jul 11 2023

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A062389, A246393, A246046, A246394, A246395.

z = 400; f[x_] := Cos[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &]  (* A062389 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &]  (* A246393 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &]  (* A246394 *)
Select[Range[0, z], f[#]*f[# + 1] > 0 &]  (* A246046 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &]  (* A246395 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &]  (* A246396 *)
SequencePosition[Table[If[Cos[k]<=0,1,0],{k,200}],{1,1}][[;;,1]] (* Harvey P. Dale, Jul 11 2023 *)

A257984 Nonhomogeneous Beatty sequence: ceiling((n - 1/2)*Pi).

Original entry on oeis.org

2, 5, 8, 11, 15, 18, 21, 24, 27, 30, 33, 37, 40, 43, 46, 49, 52, 55, 59, 62, 65, 68, 71, 74, 77, 81, 84, 87, 90, 93, 96, 99, 103, 106, 109, 112, 115, 118, 121, 125, 128, 131, 134, 137, 140, 143, 147, 150, 153, 156, 159, 162, 165, 169, 172, 175, 178, 181, 184

Offset: 1

Clark Kimberling, Jun 15 2015

Let r = Pi, s = r/(r-1), and t = 1/2. Let R be the ordered set {floor[(n + t)*r] : n is an integer} and let S be the ordered set {floor[(n - t)*s : n is an integer}; thus,
R = (..., -10, -9, -7, -6, -4, -3, -1, 0, 2, 3, 5, 6, 8, ...);
S = (..., -15, -11, -8, -5, -2, 1, 4, 7, 10, 14, 17, 20, ...).
By Fraenkel's theorem (Theorem XI in the cited paper); R and S partition the integers.
R is the set of integers n such that (cos n)*(cos(n + 1)) < 0;
S is the set of integers n such that (cos n)*(cos(n + 1)) > 0.
A246046 = (2,3,5,6,8,...), positive terms of R;
A062389 = (1,4,7,10,14,17,...), positive terms of S;
A258048 = (1,3,4,6,7,9,10,...), - (negative terms of R);
A257984 = (2,5,8,11,15,...), - (negative terms of S).
A062389 and A246046 partition the positive integers, and A258048 and A257984 partition the positive integers.

Clark Kimberling, Table of n, a(n) for n = 1..10000
A. S. Fraenkel, The bracket function and complementary sets of integers, Canadian J. of Math. 21 (1969) 6-27.

Cf. A258048 (complement), A246046, A062380, A258833.

Table[Ceiling[(n - 1/2) Pi], {n, 1, 120}] (* A257984 *)
Table[Ceiling[(n + 1/2) Pi/(Pi - 1)], {n, 0, 120}]  (* A258048 *)

a(n) = ceiling((n - 1/2)*Pi).

A258048 Nonhomogeneous Beatty sequence: a(n) = ceiling((n + 1/2)*Pi/(Pi - 1)).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97

Offset: 0

Clark Kimberling, Jun 15 2015

See A257984.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Aviezri S. Fraenkel, The bracket function and complementary sets of integers, Canadian J. of Math. 21 (1969) 6-27.

Cf. A257984 (complement), A246046, A062380, A258833.

Table[Ceiling[(n - 1/2) Pi], {n, 1, 120}] (* A257984 *)
Table[Ceiling[(n + 1/2) Pi/(Pi - 1)], {n, 0, 120}]  (* A258048 *)

a(n) = ceiling((n + 1/2)*Pi/(Pi - 1)).

cp's OEIS Frontend

A062389 a(n) = floor( (2n-1)*Pi/2 ).

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A246393 Nonnegative integers k satisfying cos(k) >= 0 and cos(k+1) <= 0.

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A246394 Nonnegative integers k satisfying cos(k) <= 0 and cos(k+1) >= 0.

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A246395 Nonnegative integers k satisfying cos(k) >= 0 and cos(k+1) >= 0.

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A246396 Nonnegative integers k satisfying cos(k) <= 0 and cos(k+1) <= 0.

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A257984 Nonhomogeneous Beatty sequence: ceiling((n - 1/2)*Pi).

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A258048 Nonhomogeneous Beatty sequence: a(n) = ceiling((n + 1/2)*Pi/(Pi - 1)).

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