A032615 -id:A032615 - cp's OEIS frontend

A000493 a(n) = floor(sin(n)).

0, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0

Offset: 0

N. J. A. Sloane

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A032615, A126564 (even bisection), A000480 (floor cos(n)).

[Floor(Sin(n)): n in [0..100]]; // Vincenzo Librandi, Jun 15 2015

Maple
```
f := n->floor(evalf(sin(n)));
```

f[ n_ ] := Floor[ N[ Sin[ n ] ] ]
Floor[Sin[Range[0,90]]] (* Harvey P. Dale, Dec 04 2012 *)

a(n) = -(A032615(n) mod 2). - Robert Israel, Jun 14 2015

A070747 a(n) = signum(sin(n)), where signum=A057427.

Original entry on oeis.org

0, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1

Offset: 0

Reinhard Zumkeller, May 04 2002

a(n) <> 0 for n>0.

			For n=10: sin(10) = sin(10-2*Pi) < 0, as Pi < 10-2*Pi < 2*Pi, therefore a(10) = signum(sin(10)) = -1.

Cf. A000493, A000494, A032615, A070748, A070751, A070752.

a(n)=(-1)^(n/Pi%2\1) \\ Charles R Greathouse IV, Jun 13 2013

a(n) = if (n==0, 0, (-1)^(n\Pi%2)); \\ Michel Marcus, Mar 20 2021

a(n) = (-1)^A032615(n) for n>0. - Michel Marcus, Mar 20 2021

A062300 a(n) = floor(cosec(Pi/(n+1))).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25

Offset: 1

Amarnath Murthy, Jun 19 2001

cosec = 1/sin. - Kevin Ryde observes that this sequences is up to the offset almost identical to A032615(n) = floor(n/Pi): the values differ after n=6 for the first time again at n=80143857. Robert Israel shows that we can nonetheless expect infinitely many differences. See the posts on the SeqFan list for details. - M. F. Hasler, Oct 19 2016

			a(99) = 31 as cosec{Pi/100} =31.8362252090976229556628738787913...

Harry J. Smith, Table of n, a(n) for n=1..1000
R. Israel, in reply to K. Ryde, Re: nearly equal floor(n/Pi) A032615 and A062300, SeqFan list, Oct 19 2016

Cf. A032615.

v=vector(150,n,floor(1/sin(Pi/(n+1)))) \\ Warning: for n=5 this may yield an incorrect value of 1 instead of a(n)=2, depending on default(realprecision).

{ default(realprecision, 50); for (n=1, 1000, write("b062300.txt", n, " ", floor(1/sin(Pi/(n+1)))) ) } \\ Harry J. Smith, Aug 04 2009

A062300(n,e=.1^precision(.1))=1\sin(Pi/(n+1+e)) \\ M. F. Hasler, Oct 19 2016

More terms from Jason Earls, Jun 22 2001

A081550 Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < Pi.

Original entry on oeis.org

6, 0, 0, 7, 8, 7, 4, 0, 1, 5, 7, 4, 8, 0, 3, 1, 4, 9, 6, 0, 6, 2, 9, 9, 2, 1, 2, 5, 9, 8, 4, 2, 5, 1, 8, 7, 1, 4, 4, 9, 1, 9, 9, 6, 5, 2, 9, 2, 6, 6, 9, 7, 1, 6, 8, 8, 3, 2, 6, 0, 7, 6, 1, 7, 7, 6, 7, 4, 3, 2, 8, 6, 9, 3, 7, 1, 5, 0, 5, 7, 5, 9, 4, 2, 2, 6, 1, 5, 0, 8, 9, 0, 4, 8, 0, 9, 4, 5, 9, 1, 5, 6, 9, 0, 1

Offset: 1

Benoit Cloitre, Apr 21 2003

			6.007874015...

Kevin O'Bryant, A generating function technique for Beatty sequences and other step sequences, Journal of Number Theory, Volume 94, Issue 2, June 2002, Pages 299-319.

Cf. A000796 (Pi).

Cf. A081544, A081564, A081573.

With[{digmax = 120}, RealDigits[Sum[1/2^Floor[k/Pi], {k, 1, 20*digmax}], 10, digmax][[1]]] (* Amiram Eldar, May 25 2023 *)

Equals Sum_{k>=1} (1/2)^floor(k/Pi) = Sum_{k>=1} 1/2^A032615(k).

Data corrected by Amiram Eldar, May 25 2023

A032616 a(n) = floor(n^2/Pi).

Original entry on oeis.org

0, 0, 1, 2, 5, 7, 11, 15, 20, 25, 31, 38, 45, 53, 62, 71, 81, 91, 103, 114, 127, 140, 154, 168, 183, 198, 215, 232, 249, 267, 286, 305, 325, 346, 367, 389, 412, 435, 459, 484, 509, 535, 561, 588, 616, 644, 673, 703, 733, 764, 795, 827, 860, 894

Offset: 0

Patrick De Geest, May 15 1998

Cf. A032615.

a[n_]:=Floor[(n^2)/Pi]; (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)

a(n) = n^2\Pi; \\ Michel Marcus, Jan 30 2022

from math import pi
print([int(n**2/pi) for n in range(54)]) # Karl-Heinz Hofmann, Feb 01 2022

a(n) = A032615(n^2). - Michel Marcus, Jan 30 2022

A082964 a(n) = round(n/Pi), n divided by Pi rounded to the nearest integer.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25

Offset: 0

Cino Hilliard, May 27 2003

			For n = 11, a(11) = (arctan(tan(11))-11)/Pi = -4;
For n = 12, a(12) = (arctan(tan(12))-12)/Pi = -4;
For n = 13, a(13) = (arctan(tan(13))-13)/Pi = -4;
For n = 14, a(14) = (arctan(tan(14))-14)/Pi = -4.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
Xavier Gordon and Pascal Sebah, Mathematical Constants and Computations
Cino Hilliard, Qfloat BCX arctan(tan(-3)) etc

Cf. A032615, A296357, A000796, A002485, A022853.

PARI
```
a(n) = round (n/Pi)
```

Simplified NAME from Hugo Pfoertner, Jul 20 2023

A296357 a(n) = ceiling of n/Pi.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27

Offset: 1

N. J. A. Sloane, Dec 15 2017

nonn

The problem asks if a(n) is also equal to ceiling(cosec(Pi/n)) for n>3.

First differs from ceiling(cosec(Pi/n)) for n>3 at n=80143857 (Stadler, 2019; Velleman and Wagon, 2020). - Amiram Eldar, Nov 08 2020

Daniel J. Velleman and Stan Wagon, Bicycle or Unicycle?, MAA Press, 2020, pp. 32 and 192-194.

Jonathan D. Lee and Stan Wagon, Proposers, Problem 12006, The American Mathematical Monthly, Vol. 124, No. 10 (2017), p. 970.
Albert Stadler and others, A Suspicious Formula Involving Pi, solution to Problem 12006, The American Mathematical Monthly, Vol. 126, No. 5 (2019), pp. 475-476.

Cf. A032615.

a[n_] := Ceiling[n/Pi]; Array[a, 100] (* Amiram Eldar, Nov 08 2020 *)

a(n)=n\Pi+1 \\ Charles R Greathouse IV, Mar 04 2018

cp's OEIS Frontend

A000493 a(n) = floor(sin(n)).

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A070747 a(n) = signum(sin(n)), where signum=A057427.

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A062300 a(n) = floor(cosec(Pi/(n+1))).

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A081550 Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < Pi.

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A032616 a(n) = floor(n^2/Pi).

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A082964 a(n) = round(n/Pi), n divided by Pi rounded to the nearest integer.

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PARI

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A296357 a(n) = ceiling of n/Pi.

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