A054386 -id:A054386 - cp's OEIS frontend

A022844 a(n) = floor(n*Pi).

0, 3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78, 81, 84, 87, 91, 94, 97, 100, 103, 106, 109, 113, 116, 119, 122, 125, 128, 131, 135, 138, 141, 144, 147, 150, 153, 157, 160, 163, 166, 169, 172, 175, 179, 182, 185, 188, 191, 194

Offset: 0

Clark Kimberling

nonn

Beatty sequence for Pi.
Differs from A127451 first at a(57). - L. Edson Jeffery, Dec 01 2013
These are the nonnegative integers m satisfying sin(m)*sin(m+1) <= 0. In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying sin(m*x)*sin((m+1)*x) <= 0, where x = Pi/r. Thus the numbers m satisfying sin(m*x)*sin((m+1)*x) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 21 2014
This can also be stated in terms of the tangent function. These are the nonnegative integers m such that tan(m/2)*tan(m/2+1/2) <= 0. In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying tan(m*x/2)*tan((m+1)*x/2) <= 0, where x = Pi/r. Thus the numbers m satisfying tan(m*x/2)*tan((m+1)*x/2) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 22 2014

			a(7)=21 because 7*Pi=21.9911... and a(8)=25 because 8*Pi=25.1327.... a(100000)=314159 because Pi=3.141592...

Ivan Panchenko, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A000796, A054386, A038130, A108591, A127451, A140758.

First differences give A063438.

R:=RieldField(10); [Floor(n*Pi(R)): n in [0..80]]; // G. C. Greubel, Sep 28 2018

a:=n->floor(n*Pi): seq(a(n),n=0..70); # Muniru A Asiru, Sep 28 2018

Floor[Pi Range[0,200]] (* Harvey P. Dale, Aug 27 2024 *)

vector(80, n, n--; floor(n*Pi)) \\ G. C. Greubel, Sep 28 2018

a(n)/n converges to Pi because |a(n)/n - Pi| = |a(n) - n*Pi|/n < 1/n. - Hieronymus Fischer, Jan 22 2006

Previous Mathematica program replaced by Harvey P. Dale, Aug 27 2024

A108591 Self-inverse integer permutation induced by Beatty sequences for Pi and Pi/(Pi-1).

Original entry on oeis.org

3, 6, 1, 9, 12, 2, 15, 18, 4, 21, 25, 5, 28, 31, 7, 34, 37, 8, 40, 43, 10, 47, 50, 53, 11, 56, 59, 13, 62, 65, 14, 69, 72, 16, 75, 78, 17, 81, 84, 19, 87, 91, 20, 94, 97, 100, 22, 103, 106, 23, 109, 113, 24, 116, 119, 26, 122, 125, 27, 128, 131, 29, 135, 138, 30, 141, 144

Offset: 1

Reinhard Zumkeller, Jun 11 2005

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to Beatty sequences

Cf. A108590, A108592, A108599.

a(A022844(n))=A054386(n) and a(A054386(n))=A022844(n).

a(53)/a(54) joined by Georg Fischer, May 24 2022

A108586 Floor(2nPi/(2*Pi-1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset: 1

Reinhard Zumkeller, Jun 11 2005

nonn

Beatty sequence for 2*Pi/(2*Pi-1); complement of A038130; not the same as A108120: a(37)=44 <> A108120(37)=43.

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A054386, A140758, A108592.

With[{c=2Pi},Floor[(c*Range[80])/(c-1)]] (* Harvey P. Dale, Apr 21 2024 *)

A108589 a(n) = floor(n*Pi/(Pi-2)).

Original entry on oeis.org

2, 5, 8, 11, 13, 16, 19, 22, 24, 27, 30, 33, 35, 38, 41, 44, 46, 49, 52, 55, 57, 60, 63, 66, 68, 71, 74, 77, 79, 82, 85, 88, 90, 93, 96, 99, 101, 104, 107, 110, 112, 115, 118, 121, 123, 126, 129, 132, 134, 137, 140, 143, 145, 148, 151, 154, 156, 159, 162, 165, 167

Offset: 1

Reinhard Zumkeller, Jun 11 2005

Beatty sequence for Pi/(Pi-2); complement of A140758.

Vincenzo Librandi, Table of n, a(n) for n = 1..7000
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A054386, A108586, A108590, A140758.

R:= RealField(40); [Floor(n*Pi(R)/(Pi(R)-2)): n in [1..60]]; // G. C. Greubel, Oct 21 2023

A108589:=n->floor(n*Pi/(Pi-2)); seq(A108589(n), n=1..50); # Wesley Ivan Hurt, Apr 19 2014

With[{c=Pi/(Pi-2)},Floor[c*Range[70]]] (* Harvey P. Dale, Apr 19 2014 *)

[floor(n*pi/(pi-2)) for n in range(1,61)] # G. C. Greubel, Oct 21 2023

A246388 Nonnegative integers k satisfying sin(k) >= 0 and sin(k+1) <= 0.

Original entry on oeis.org

3, 9, 15, 21, 28, 34, 40, 47, 53, 59, 65, 72, 78, 84, 91, 97, 103, 109, 116, 122, 128, 135, 141, 147, 153, 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: 0

Clark Kimberling, Aug 24 2014

A246388 and A038130 (Beatty sequence for 2*Pi) partition A022844 (Beatty sequence for Pi). Likewise, A054386, the complement of A022844, is partitioned by A246389 and A246390. (See the Mathematica program.)

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

Cf. A022844, A038130, A054386, A246389, A246390.

z = 400; f[x_] := Sin[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &] (* A022844 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &]  (* A246388 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &] (* A038130 *)
Select[Range[0, z], f[#]*f[# + 1] > 0 &] (* A054386 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &]  (* A246389 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &] (* A246390 *)

A115239 a(1) = floor(Pi) = 3; a(n+1) = floor(a(n)*Pi).

Original entry on oeis.org

3, 9, 28, 87, 273, 857, 2692, 8457, 26568, 83465, 262213, 823766, 2587937, 8130243, 25541911, 80242279, 252088554, 791959549, 2488014301, 7816327450, 24555716894, 77144059797, 242355211526, 761381352089, 2391950062303

Offset: 1

Hieronymus Fischer, Jan 17 2006

nonn

a(n+1)/a(n) converges to Pi. Similar to sequence A085839 but with a simpler definition.

Subset of the Beatty sequence of Pi = A022844 = floor(n*Pi). Primes in this sequence include a(1) = 3, a(6) = 857, a(15) = 25541911. - Jonathan Vos Post, Jan 18 2006

			a(2) = floor(a(1)*Pi) = floor(3*Pi) = 9;
a(3) = floor(a(2)*Pi) = floor(9*Pi) = 28;
a(4) = floor(a(3)*Pi) = floor(28*Pi) = 87.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Beatty Sequence.

Cf. A085839.

Cf. A022844, A038130, A054386, A108591.

A[1]:= 3:
for n from 2 to 50 do A[n]:= floor(Pi*A[n-1]) od:
seq(A[i],i=1..50); # Robert Israel, Feb 07 2016

a[1] = Floor[Pi]; a[n_] := a[n] = Floor[a[n - 1]*Pi]; Array[a, 25] (* Robert G. Wilson v, Jan 18 2006 *)
NestList[Floor[Pi #]&,3,30] (* Harvey P. Dale, Mar 30 2012 *)

More terms from Robert G. Wilson v, Jan 18 2006

A246389 Nonnegative integers k satisfying sin(k) >= 0 and sin(k+1) >= 0.

Original entry on oeis.org

0, 1, 2, 7, 8, 13, 14, 19, 20, 26, 27, 32, 33, 38, 39, 44, 45, 46, 51, 52, 57, 58, 63, 64, 70, 71, 76, 77, 82, 83, 88, 89, 90, 95, 96, 101, 102, 107, 108, 114, 115, 120, 121, 126, 127, 132, 133, 134, 139, 140, 145, 146, 151, 152, 158, 159, 164, 165, 170, 171

Offset: 0

Clark Kimberling, Aug 24 2014

A246388 and A038130 (Beatty sequence for 2*Pi) partition A022844 (Beatty sequence for Pi). Likewise, A054386, the complement of A022844, is partitioned by A246389 and A246390. (See the Mathematica program.)

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

Cf. A022844, A038130, A054386, A246388, A246390.

Digits := 100:
isA246389 := proc(k)
    if evalf(sin(k)) >= 0 and evalf(sin(k+1)) >= 0 then
        return true ;
    else
        return false ;
    end if;
end proc:
A246389 := proc(n)
    option remember ;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA246389(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A246389(n),n=1..100) ; # assumes offset 1 R. J. Mathar, Jan 18 2024

z = 400; f[x_] := Sin[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &] (* A022844 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &]  (* A246388 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &] (* A038130 *)
Select[Range[0, z], f[#]*f[# + 1] > 0 &] (* A054386 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &]  (* A246389 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &] (* A246390 *)

A127450 Beatty sequence for 1/(e^Pi - Pi^e), complement of A127451.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102

Offset: 1

Robert G. Wilson v, Jan 14 2007

Differs from A054386 at term n=122, where A054386(122)=178, A127450(122)=179. - Martin Fuller, May 10 2007

Martin Fuller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A063504, A054386, A059564.

Table[Floor[n/(Exp[Pi] - Pi^E)], {n, 70}]

a(n) = floor(n/(e^Pi - Pi^e))

Definition corrected by N. J. A. Sloane, May 10 2007

A246390 Nonnegative integers k satisfying sin(k) <= 0 and sin(k+1) <= 0.

Original entry on oeis.org

4, 5, 10, 11, 16, 17, 22, 23, 24, 29, 30, 35, 36, 41, 42, 48, 49, 54, 55, 60, 61, 66, 67, 68, 73, 74, 79, 80, 85, 86, 92, 93, 98, 99, 104, 105, 110, 111, 112, 117, 118, 123, 124, 129, 130, 136, 137, 142, 143, 148, 149, 154, 155, 156, 161, 162, 167, 168, 173

Offset: 0

Clark Kimberling, Aug 24 2014

A246388 and A038130 (Beatty sequence for 2*Pi) partition A022844 (Beatty sequence for Pi). Likewise, A054386, the complement of A022844, is partitioned by A246389 and A246390. (See the Mathematica program.)

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

Cf. A022844, A038130, A054386, A246388, A246389.

z = 400; f[x_] := Sin[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &] (* A022844 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &]  (* A246388 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &] (* A038130 *)
Select[Range[0, z], f[#]*f[# + 1] > 0 &] (* A054386 *)
Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &]  (* A246389 *)
Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &] (* A246390 *)
SequencePosition[Table[If[Sin[n]<=0,1,0],{n,200}],{1,1}][[;;,1]] (* Harvey P. Dale, Apr 02 2023 *)

cp's OEIS Frontend

A022844 a(n) = floor(n*Pi).

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A108591 Self-inverse integer permutation induced by Beatty sequences for Pi and Pi/(Pi-1).

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A108586 Floor(2*n*Pi/(2*Pi-1)).

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A108589 a(n) = floor(n*Pi/(Pi-2)).

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A246388 Nonnegative integers k satisfying sin(k) >= 0 and sin(k+1) <= 0.

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A115239 a(1) = floor(Pi) = 3; a(n+1) = floor(a(n)*Pi).

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Maple

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A246389 Nonnegative integers k satisfying sin(k) >= 0 and sin(k+1) >= 0.

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Maple

Mathematica

A127450 Beatty sequence for 1/(e^Pi - Pi^e), complement of A127451.

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A108586 Floor(2nPi/(2*Pi-1)).