A049469 -id:A049469 - cp's OEIS frontend

A108587 a(n) = floor(n/(1-sin(1))).

6, 12, 18, 25, 31, 37, 44, 50, 56, 63, 69, 75, 82, 88, 94, 100, 107, 113, 119, 126, 132, 138, 145, 151, 157, 164, 170, 176, 182, 189, 195, 201, 208, 214, 220, 227, 233, 239, 246, 252, 258, 264, 271, 277, 283, 290, 296, 302, 309, 315, 321, 328, 334, 340, 346

Offset: 1

Reinhard Zumkeller, Jun 11 2005

nonn

Beatty sequence for 1/(1-sin(1)); complement of A108120.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A038130, A049469, A108120, A108593, A206530.

[Floor(n/(1-Sin(1))): n in [1..60]]; // G. C. Greubel, Dec 19 2022

Table[Floor[n/(1-Sin[1])], {n, 60}] (* G. C. Greubel, Dec 19 2022 *)

[int(n/(1-sin(1))) for n in range(1,61)] # G. C. Greubel, Dec 19 2022

A206530 Decimal expansion of 1/(1-sin(1)).

Original entry on oeis.org

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

Offset: 1

Seiichi Kirikami, Feb 11 2012

The value of the limit of (A206307+6*A206308) / (A206308).

			6.3079935164437400275135217398...

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002943, A049469, A125202, A206307, A206308.

SetDefaultRealField(RealField(150)); 1/(1-Sin(1)); // G. C. Greubel, Dec 20 2022

Mathematica
```
RealDigits[N[1/(1-Sin[1]), 150]][[1]]
```

numerical_approx(1/(1-sin(1)), digits=150) # G. C. Greubel, Dec 20 2022

Equals 1/(1-A049469).
A206307/A206308 + 6 -> 1/(1-A049469).
Abs(A206308/(1-sin(1)) - (A206307 + 6*A206308)) -> 0.

A334363 Decimal expansion of Sum_{k>=0} 1/(4*k+1)!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 24 2020

			1/1! + 1/5! + 1/9! + ... = 1.008336089225848981767442...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 36.

Cf. A001113, A049469, A073742, A143820, A332890, A334364, A334365.

RealDigits[(Sin[1] + Sinh[1])/2, 10, 110] [[1]]

suminf(k=0, 1/(4*k+1)!) \\ Michel Marcus, Apr 25 2020

Equals (sin(1) + sinh(1))/2.

A334365 Decimal expansion of Sum_{k>=0} 1/(4*k+3)!.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

			1/3! + 1/7! + 1/11! + ... = 0.1668651044179524751...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.
Index entries for transcendental numbers.

Cf. A001113, A049469, A073742, A332890, A334363, A334364.

RealDigits[(Sinh[1] - Sin[1])/2, 10, 111] [[1]]

suminf(k=0, 1/(4*k+3)!) \\ Michel Marcus, Apr 25 2020

Equals (sinh(1) - sin(1))/2.

A067919 Engel expansion of sin(1).

Original entry on oeis.org

2, 2, 3, 11, 14, 27, 28, 66, 212, 231, 552, 2842, 3774, 6038, 6784, 10950, 32948, 78591, 97875, 98342, 123569, 139837, 159698, 1102838, 3256476, 20329622, 34385124, 60999878, 82669919, 85820365, 389915995, 4274338879, 18907353107, 62875944378, 74931184173

Offset: 1

Benoit Cloitre, Mar 03 2002

			sin(1) = 0.84147... = A049469 has the Engel expansion 1/2 + 1/(2*2) + 1/(2*2*3) + ...

See A006784 for explanation of Engel expansions.

Cf. A049469, A084651, A084652.

EngelExp[A_,n_]:=Join[Array[1&,Floor[A]],First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]],Expand[ #[[1]]#[[2]]-1]}&,{Ceiling[1/(A-Floor[A])],A-Floor[A]},n-1]]; EngelExp[N[Sin[1],6! ],50] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2009 *)

s=sin(1); for(i=1,30,s=s*ceil(1/s)-1; print1(ceil(1/s),","); );

a(1) inserted by Hauke Worpel (hw1(AT)email.com), Jun 01 2003

Edited by N. J. A. Sloane, Nov 01 2008 at the suggestion of R. J. Mathar

A233383 Decimal expansion of the absolute value of Sum_{n>=1} (-1)^n*sin(1/n).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 08 2013

If the contribution of the first term, -sin(1) = -A049469, is omitted, the constant becomes Sum_{n>=1} (sin(1/(2n)) - sin(1/(2n+1))) = 0.29067413667396703114243536419518058522...

			0.550796848133929475510066957...

oldrinb, Evaluating the infinite series sum_n=1^infty [sin(1/2n)-sin(1/(2n+1))], math.stackexchange, Aug 22 2013

M := 141 :
Digits := 120 :
s := sin(1/2/n)-sin(1/(2*n+1)) :
add(subs(n=i,s),i=1..M) :
pre := evalf(%) :
zetaM := proc(s,M)
    local n ;
    Zeta(s)-add(1/n^s,n=1..M) ;
    evalf(%) ;
end proc:
for dd from 75 to 90 by 5 do
    subs(n=1/eps,s) ;
    taylor(%,eps=0,dd+1) ;
    t := gfun[seriestolist](%,'ogf') ;
    add( op(j,t)*zetaM(j-1,M),j=3..nops(t)) ;
    x := pre+% ;
    print(x) ;
end do:
# now sum_{n>=1} (-1)^n*sin(1/n) = -0.5570986.
x-sin(1.0) ;

digits = 105; NSum[(-1)^n*Sin[1/n], {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+10] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 24 2014 *)

A280188 Base-60 (Babylonian or sexagesimal) expansion of sine of 1 degree.

Original entry on oeis.org

1, 2, 49, 43, 11, 14, 44, 16, 26, 18, 28, 49, 20, 26, 50, 41, 13, 6, 46, 25, 26, 26, 34, 6, 40, 18, 50, 31, 6, 35, 20, 44, 6, 39, 18, 5, 38, 58, 2, 0, 5, 4, 33, 59, 11, 35, 33, 50, 34, 7, 56, 43, 38, 30, 15, 49, 36, 42, 6, 43, 10, 38, 45, 53, 15, 59, 7, 19, 46

Offset: 1

Mohammad K. Azarian, Dec 28 2016

The Fifteenth Century Persian mathematician Jamshid Al-Kashi was the first to calculate the value of sine of one degree correct to ten sexagesimal places (17 decimal digits) in his Risala al-Watar wa'l Jaib.

Mohammad K. Azarian, A Study of Risala al-Watar wa'l Jaib ("The Treatise on the Chord and Sine"), Forum Geometricorum, Volume 15 (2015) 229-242. Mathematical Reviews, MR 3418854 (Reviewed), Zentralblatt MATH, Zbl 1328.01015.

Cf. A019810, A019812, A049469, A110937, A280189.

Mathematica
```
RealDigits[Sin[1 Degree], 60, 200][[1]]
```

More terms from Jon E. Schoenfield, Jan 13 2017

A372339 Decimal expansion of Sum_{k>=0} (-1)^k * (k+2) / ((k+1)(2k)!).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 28 2024

			1.3038488872202121655078144655895278104420573...

Cf. A049469, A049470, A372338.

s = Sum[(-1)^k (k + 2)/((k + 1) (2 k)!), {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d]]

Equals -2 + 3*cos(1) + 2*sin(1).

A091033 Third column (k=4) of array A090438 ((4,2)-Stirling2).

Original entry on oeis.org

1, 180, 25200, 4233600, 898128000, 239740300800, 79332244992000, 32011868528640000, 15509750302126080000, 8898339094906060800000, 5971815866682429603840000, 4637851802955964809216000000

Offset: 2

Wolfdieter Lang, Jan 23 2004

Cf. A091032 (second column of A090438 divided by 8), A091034 (fourth column divided by 24), A000384, A090438.

Cf. A001620, A049469, A049470, A073742, A099281, A099282, A099283, A099284.

a[n_] := (n-1)*(2*n-3)*(2*n)!/4!; Array[a, 12, 2] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(2*n-3)*(2*n)!/4!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 4), n>=2.
a(n) = (n-1)*(2*n-3)*(2*n)!/4! = binomial(2*(n-1), 2)*(2*n)!/4! = A000384(n-1)*(2*n)!/4!, n>=2.
E.g.f.: (6*hypergeom([1/2, 1], [], 4*x) - 4*hypergeom([1, 3/2], [], 4*x) + hypergeom([3/2, 2], [], 4*x) -3)/4! (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=2} 1/a(n) = -20 + 24*Gamma - 16*CoshIntegral(1) + 16*sinh(1) + 8*SinhIntegral(1).
Sum_{n>=2} (-1)^n/a(n) = 4 - 24*gamma + 16*cos(1) + 24*CosIntegral(1) - 16*sin(1) + 8*SinIntegral(1). (End)

A091034 Fourth column (k=5) of array A090438 ((4,2)-Stirling2) divided by 24.

Original entry on oeis.org

1, 280, 70560, 19958400, 6659452800, 2644408166400, 1244905998336000, 689322235650048000, 444916954745303040000, 331767548149023866880000, 283424276847308960563200000, 275246422218908346286080000000

Offset: 3

Wolfdieter Lang, Jan 23 2004

Cf. A091033 (third column of A090438), A091035 (fifth column), A090438.

Cf. A001620, A049469, A049470, A073742, A073743, A099281, A099282, A099283, A099284.

a[n_] := (n - 1)*(n - 2)*(2*n - 3)*(2*n)!/(5!*(3!)^2); Array[a, 12, 3] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(n-2)*(2*n-3)*(2*n)!/(5!*(3!)^2); \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 5)/24, n>=3.
a(n) = (n-1)*(n-2)*(2*n-3)*(2*n)!/(5!*(3!)^2), n>=3.
E.g.f.: (Sum_{p=2..5} (((-1)^(p+1))*binomial(5, p)*hypergeom([(p-1)/2, p/2], [], 4*x)) + 4)/(5!*4!) (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=3} 1/a(n) = 2010 - 4680*Gamma + 1800*cosh(1) + 4680*CoshIntegral(1) - 2520*sinh(1) - 2880*SinhIntegral(1).
Sum_{n>=3} (-1)^(n+1)/a(n) = -2010 - 3960*gamma + 3240*cos(1) + 3960*CosIntegral(1) - 1800*sin(1) + 2880*SinIntegral(1). (End)

cp's OEIS Frontend

A108587 a(n) = floor(n/(1-sin(1))).

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Magma

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A206530 Decimal expansion of 1/(1-sin(1)).

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A334363 Decimal expansion of Sum_{k>=0} 1/(4*k+1)!.

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A334365 Decimal expansion of Sum_{k>=0} 1/(4*k+3)!.

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A067919 Engel expansion of sin(1).

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A233383 Decimal expansion of the absolute value of Sum_{n>=1} (-1)^n*sin(1/n).

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Maple

Mathematica

A280188 Base-60 (Babylonian or sexagesimal) expansion of sine of 1 degree.

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Mathematica

Extensions

A372339 Decimal expansion of Sum_{k>=0} (-1)^k * (k+2) / ((k+1)*(2*k)!).

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A372339 Decimal expansion of Sum_{k>=0} (-1)^k * (k+2) / ((k+1)(2k)!).