A108591 -id:A108591 - 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

A054386 Beatty sequence for Pi/(Pi-1); complement of A022844.

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

Eric W. Weisstein

nonn

Differs from A127450 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. A022844, A108586, A108589, A108591.

R:=RealField(30); [Floor(n*Pi(R)/(Pi(R)-1)): n in [1..80]]; // G. C. Greubel, Oct 22 2023

Floor[Pi*Range[80]/(Pi-1)] (* G. C. Greubel, Oct 22 2023 *)

[floor(n*pi/(pi-1)) for n in range(1,81)] # G. C. Greubel, Oct 22 2023

A108592 Self-inverse integer permutation induced by Beatty sequences for 2Pi and 2Pi/(2*Pi-1).

Original entry on oeis.org

6, 12, 18, 25, 31, 1, 37, 43, 50, 56, 62, 2, 69, 75, 81, 87, 94, 3, 100, 106, 113, 119, 125, 131, 4, 138, 144, 150, 157, 163, 5, 169, 175, 182, 188, 194, 7, 201, 207, 213, 219, 226, 8, 232, 238, 245, 251, 257, 263, 9, 270, 276, 282, 289, 295, 10, 301, 307, 314, 320, 326, 11, 333, 339, 345, 351, 358, 364, 13, 370

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, A108591, A108593.

a30(n) = floor(n*2*Pi);
a86(n) = floor(2*n*Pi/(2*Pi-1));
lista(nn) = {my(vb = vector(nn, n, a30(n))); my(vc = vector(nn, n, a86(n))); my(va = vector(nn)); for (n=1, nn, if (vb[n] <= nn, va[vb[n]] = vc[n]); if (vc[n] <= nn, va[vc[n]] = vb[n]);); va;} \\ Michel Marcus, May 25 2022

a(A038130(n))=A108586(n) and a(A108586(n))=A038130(n).

Four terms corrected by Georg Fischer and Michel Marcus, May 25 2022

A108590 Self-inverse integer permutation induced by Beatty sequences for Pi/2 and Pi/(Pi-2).

Original entry on oeis.org

2, 1, 5, 8, 3, 11, 13, 4, 16, 19, 6, 22, 7, 24, 27, 9, 30, 33, 10, 35, 38, 12, 41, 14, 44, 46, 15, 49, 52, 17, 55, 57, 18, 60, 20, 63, 66, 21, 68, 71, 23, 74, 77, 25, 79, 26, 82, 85, 28, 88, 90, 29, 93, 96, 31, 99, 32, 101, 104, 34, 107, 110, 36, 112, 115, 37, 118, 39, 121

Offset: 1

Reinhard Zumkeller, Jun 11 2005

nonn

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. A108591, A108592.

a(A140758(n))=A108589(n) and a(A108589(n))=A140758(n).

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

A108599 Self-inverse integer permutation induced by Beatty sequences for e and e/(e-1).

Original entry on oeis.org

2, 1, 5, 8, 3, 10, 13, 4, 16, 6, 19, 21, 7, 24, 27, 9, 29, 32, 11, 35, 12, 38, 40, 14, 43, 46, 15, 48, 17, 51, 54, 18, 57, 59, 20, 62, 65, 22, 67, 23, 70, 73, 25, 76, 78, 26, 81, 28, 84, 86, 30, 89, 92, 31, 95, 97, 33, 100, 34, 103, 106, 36, 108, 111, 37, 114, 39, 116, 119, 41

Offset: 1

Reinhard Zumkeller, Jun 11 2005

nonn

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. A108591.

a(A022843(n))=A054385(n) and a(A054385(n))=A022843(n).

A110119 Self-inverse integer permutation induced by Beatty sequences for x and (x+1)/(2*sqrt(2)) with x=sqrt(2)+sqrt(3).

Original entry on oeis.org

3, 6, 1, 9, 12, 2, 15, 18, 4, 22, 25, 5, 28, 31, 7, 34, 37, 8, 40, 44, 47, 10, 50, 53, 11, 56, 59, 13, 62, 66, 14, 69, 72, 16, 75, 78, 17, 81, 84, 19, 88, 91, 94, 20, 97, 100, 21, 103, 106, 23, 110, 113, 24, 116, 119, 26, 122, 125, 27, 128, 132, 29, 135, 138, 141, 30, 144

Offset: 1

Reinhard Zumkeller, Jul 13 2005

nonn

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

Cf. A108591, A110117, A110118.

Cf. A135611 (sqrt(2)+sqrt(3)).

a(A110117(n)) = A110118(n) and a(A110118(n)) = A110117(n).

cp's OEIS Frontend

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

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A054386 Beatty sequence for Pi/(Pi-1); complement of A022844.

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A108592 Self-inverse integer permutation induced by Beatty sequences for 2*Pi and 2*Pi/(2*Pi-1).

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A108590 Self-inverse integer permutation induced by Beatty sequences for Pi/2 and Pi/(Pi-2).

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

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Maple

Mathematica

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A108599 Self-inverse integer permutation induced by Beatty sequences for e and e/(e-1).

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A110119 Self-inverse integer permutation induced by Beatty sequences for x and (x+1)/(2*sqrt(2)) with x=sqrt(2)+sqrt(3).

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A108592 Self-inverse integer permutation induced by Beatty sequences for 2Pi and 2Pi/(2*Pi-1).