A049471 -id:A049471 - cp's OEIS frontend

A093178 If n is even then 1, otherwise n.

1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 21, 1, 23, 1, 25, 1, 27, 1, 29, 1, 31, 1, 33, 1, 35, 1, 37, 1, 39, 1, 41, 1, 43, 1, 45, 1, 47, 1, 49, 1, 51, 1, 53, 1, 55, 1, 57, 1, 59, 1, 61, 1, 63, 1, 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 75, 1, 77, 1, 79, 1, 81, 1, 83, 1, 85

Offset: 0

Michael Somos, Mar 27 2004

Continued fraction expansion for tan(1).
1 followed by run lengths of A062557 = 2n-1 1's followed by a 2. - Jeremy Gardiner, Aug 12 2012
Greatest common divisor of n and (n+1) mod 2. - Bruno Berselli, Mar 07 2017

			1.557407724654902230506974807... = 1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + ...))))
G.f. = 1 + x + x^2 + 3*x^3 + x^4 + 5*x^5 + x^6 + 7*x^7 + x^8 + 9*x^9 + x^10 + ...

Harry J. Smith, Table of n, a(n) for n = 0..20000
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
Simon Plouffe, A Search for a mathematical expression for mass ratios using a large database. page 3.
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Equals |A009001(n)|.

Cf. A133080, A049471 (decimal expansion), A009001, A161738, A062557, A124625.

A093178:=n->(n+1+(1-n)*(-1)^n)/2; seq(A093178(k), k=0..100); # Wesley Ivan Hurt, Oct 19 2013

Join[{1},Riffle[Range[1,85,2],1]] (* or *) Array[If[EvenQ[#],1,#]&,87,0] (* Harvey P. Dale, Nov 23 2011 *)

PARI
```
{a(n) = if( n%2, n, 1)};
```

G.f.: (1+x-x^2+x^3)/(1-x^2)^2.
a(n) = (-1)^n * a(-n) for all n in Z.
a(n) = (1/2) * [ 1 + n + (1-n)*(-1)^n ]. - Ralf Stephan, Dec 02 2004
a(n) = n^n mod (n+1) for n > 0. - Amarnath Murthy, Apr 18 2004
Satisfies a(0) = 1, a(n+1) = a(n) + n if a(n) < n else a(n+1) = a(n)/n. - Amarnath Murthy, Oct 29 2002
a(n) = ((n+1)+(1-n)(-1)^n)/2 and have e.g.f. (1+x)cosh(x). - Paul Barry, Apr 09 2003
a(n) = binomial(n, 2*floor(n/2)). - Paul Barry, Dec 28 2006
Starting (1, 1, 3, 1, 5, 1, 7, ...) = A133080^(-1) * [1,2,3,...]. - Gary W. Adamson, Sep 08 2007
a(n) = denom(b(n+2)/b(n+1)) with b(n) = product((2*n-3-2*k), k=0..floor(n/2-1)). - Johannes W. Meijer, Jun 18 2009
a(n) = 2*floor(n/2) - n*(n-1 mod 2) + 1. - Wesley Ivan Hurt, Oct 19 2013
a(n) = n^(n mod 2). - Wesley Ivan Hurt, Apr 16 2014

A073449 Decimal expansion of cot(1).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 01 2002

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.64209261593433070300641998659...

Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, pp. 413-414.
Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.
Index entries for transcendental numbers

Cf. A049471 (tan(1)=1/A073449), A049469 (sin(1)), A049470 (cos(1)), A073447 (csc(1)), A073448 (sec(1)).

RealDigits[Cot[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)

PARI
```
cotan(1)
```

Equals Sum_{k>=0} (-1)^k * B(2*k) * 2^(2*k) / (2*k)!, where B(k) is the k-th Bernoulli number. - Amiram Eldar, May 15 2021

A073447 Decimal expansion of csc(1).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 01 2002

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			1.18839510577812121626159945237...

Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395. Solution published in Vol. 37 , No. 5, November 2006, pp. 394-395.
Index entries for transcendental numbers.

Cf. A049469 (sin(1)=1/A073447), A049470 (cos(1)), A049471 (tan(1)), A073448 (sec(1)), A073449 (cot(1)).

Cf. A027641, A027642.

RealDigits[Csc[1], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

PARI
```
1/sin(1)
```

Equals Sum_{n=-oo..oo} ((-1)^n/(1 + n*Pi)). - Jean-François Alcover, Mar 21 2013.

Equals Sum_{k>=0} (-1)^k * (2 - 4^k) * bernoulli(2*k)/(2*k)! = Sum_{k>=0} (-1)^k * (2 - 4^k) * A027641(2*k)/(A027642(2*k)*(2*k)!). - Amiram Eldar, Aug 03 2020

A073448 Decimal expansion of sec(1).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 01 2002

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			1.85081571768092561791175324139...

Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, pp. 413-414.
Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.
Index entries for transcendental numbers

Cf. A049470 (cos(1)=1/A073448), A049469 (sin(1)), A049471 (tan(1)), A073447 (csc(1)), A073449 (cot(1)), A122045.

RealDigits[Sec[1],10,120][[1]] (* Harvey P. Dale, Mar 13 2013 *)

PARI
```
1/cos(1)
```

Equals Sum_{k>=0} (-1)^k * E(2*k) / (2*k)!, where E(k) is the k-th Euler number (A122045). - Amiram Eldar, May 15 2021

A009001 Expansion of e.g.f: (1+x)*cos(x).

Original entry on oeis.org

1, 1, -1, -3, 1, 5, -1, -7, 1, 9, -1, -11, 1, 13, -1, -15, 1, 17, -1, -19, 1, 21, -1, -23, 1, 25, -1, -27, 1, 29, -1, -31, 1, 33, -1, -35, 1, 37, -1, -39, 1, 41, -1, -43, 1, 45, -1, -47, 1, 49, -1, -51, 1, 53, -1, -55, 1, 57, -1, -59, 1, 61, -1, -63, 1, 65, -1, -67, 1, 69, -1, -71, 1, 73, -1, -75, 1, 77, -1, -79, 1, 81, -1, -83, 1, 85

Offset: 0

R. H. Hardin, N. J. A. Sloane, Simon Plouffe, David W. Wilson

If signs are ignored, continued fraction for tan(1) (cf. A093178).

			tan(1) = 1.557407724654902230... = 1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 15 2009
G.f. = 1 + x - x^2 - 3*x^3 + x^4 + 5*x^5 - x^6 - 7*x^7 + x^8 + 9*x^9 - x^10 + ...

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A009531, A049471 (decimal expansion of tan(1)).

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Cos(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 21 2018

seq(coeff(series(factorial(n)*(1+x)*cos(x), x,n+1),x,n),n=0..90); # Muniru A Asiru, Jul 21 2018

With[{nn=90},CoefficientList[Series[(1+x)Cos[x],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Jul 15 2012 *)
LinearRecurrence[{0, -2, 0, -1}, {1, 1, -1, -3}, 100] (* Jean-François Alcover, Feb 21 2020 *)
FoldList[If[Mod[#2, 4]==1, #1*#2, If[Mod[#2, 4]==2, #1-#2,If[Mod[#2,4] ==3,#1*#2,#1+#2]]]&, 1, Range[1, 85]] (* James C. McMahon, Oct 12 2023 *)

{a(n) = (-1)^(n\2) * if( n%2, n, 1)} /* Michael Somos, Oct 16 2006 */

a(n) = (-1)^(n/2) if n is even, n*(-1)^((n-1)/2) if n is odd.
a(n) = -a(n-2) if n is even, 2*a(n-1) - a(n-2) if n is odd. - Michael Somos, Jan 26 2014
From Henry Bottomley, Oct 19 2001: (Start)
a(n) = (n^n mod (n+1))*(-1)^floor(n/2) for n > 0.
a(n) = (-1)^n*(a(n-2) - a(n-1)) - a(n-3) for n > 2. (End)
G.f.: (1+x+x^2-x^3)/(1+x^2)^2.
E.g.f.: (1+x)*cos(x) = U(0) where U(k) = 1 + x - x^2/((2*k+1)*(2*k+2)) * U(k+1). - Sergei N. Gladkovskii, Oct 17 2012 [Edited by Michael Somos, Jan 26 2014]
From James C. McMahon, Oct 12 2023: (Start)
a(0) = 1; for n > 1,
a(n) = a(n-1) * n if n mod 4 = 1,
       a(n-1) - n if n mod 4 = 2,
       a(n-1) * n if n mod 4 = 3,
       a(n-1) + n if n mod 4 = 4. (End)

Formula corrected by Olivier Gérard, Mar 15 1997

Definition clarified by Harvey P. Dale, Jul 15 2012

A161011 Decimal expansion of tan(1/2).

Original entry on oeis.org

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

Offset: 0

Harry J. Smith, Jun 13 2009

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.546302489843790513255179465780285383297551720179791246164091385932907...

Harry J. Smith, Table of n, a(n) for n = 0..20000
MathOverflow, What is the effect of adding 1/2 to a continued fraction?
Wikipedia, Lindemann-Weierstrass theorem
Index entries for transcendental numbers

Cf. A019425 (continued fraction). Cf. A049471, A161011 through A161019.

RealDigits[N[Tan[1/2],6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2009 *)

default(realprecision, 20080); x=10*tan(1/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b161011.txt", n, " ", d));

From Peter Bala, Nov 17 2019: (Start)
Related simple continued fraction expansions:
tan(1/2) = [0; 1, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, ...]. See A019425.
2*tan(1/2) = [1, 10, 1, 3, 1, 26, 1, 7, 1, 42, 1, 11, 1, 58, 1, 15, 1, 74, 1, 19, 1, 90, ...]
(1/2)*tan(1/2) = [0; 3, 1, 1, 1, 18, 1, 5, 1, 34, 1, 9, 1, 50, 1, 13, 1, 66, 1, 17, 1, 82, ...].
tan(1/2)/(1 - tan(1/2)) = [1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, ...]
2*tan(1/2)/(1 - tan(1/2)) = [2, 2, 2, 4, 2, 6, 2, 8, 2, 10, 2, 12, ...]
4*tan(1/2)/(1 - tan(1/2)) = [4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 4, 7, ...]. (End)

A348140 a(n) is the numerator of tan(n * arctan(1/n)).

Original entry on oeis.org

1, 4, 13, 240, 719, 42372, 92567, 14970816, 21475201, 8825080100, 7836127861, 7809130867824, 4132643140079, 9678967816041188, 2973238691433583, 16000787866533953280, 2798084251807349761, 34017524842099233036996, 3336132453587291393821, 90417110945911655996319600

Offset: 1

Amiram Eldar, Oct 02 2021

			The fractions begin with 1, 4/3, 13/9, 240/161, 719/475, 42372/27755, 92567/60319, 14970816/9722113, 21475201/13913289, 8825080100/5707904499, ...

Amiram Eldar, Table of n, a(n) for n = 1..387

Cf. A049471, A348131, A348132, A348141 (denominators).

f[n_] := Module[{s = 1/n}, Do[s = (s + 1/n)/(1 - s/n), {k, 1, n - 1}]; s]; Numerator @ Array[f, 20]

Lim_{n->oo} a(n)/A348141(n) = tan(1) (A049471).

A009753 Expansion of tan(x)/(1+x).

Original entry on oeis.org

0, 1, -2, 8, -32, 176, -1056, 7664, -61312, 559744, -5597440, 61925632, -743107584, 9682766848, -135558735872, 2035284795392, -32564556726272, 553807329689600, -9968531934412800, 189431195638956032, -3788623912779120640

Offset: 0

R. H. Hardin

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A049471 (tan(1)).

G(x):= tan(x)/(1+x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009

CoefficientList[Series[Tan[x]/(1 + x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 24 2015 *)

x='x+O('x^30); concat([0], Vec(serlaplace(tan(x)/(1+x)))) \\ G. C. Greubel, Feb 12 2018

a(n) ~ n! * (-1)^(n+1) * tan(1). - Vaclav Kotesovec, Jan 24 2015

Extended with signs by Olivier Gérard, Mar 15 1997

A085666 Decimal expansion of -tan(tan(tan(1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 15 2003

			0.86351885487805888198701559313643185077224760581857...

Jens Blanck, Exact real arithmetic systems: results of competition, pp. 389-393 of J. Blanck et al., eds., Computability and Complexity in Analysis (CCA 2000), Lect. Notes Computer Science, Springer-Verlag, 2001; alternative link.

Cf. A049471.

RealDigits[Tan[Tan[Tan[1]]], 10, 120][[1]] (* Amiram Eldar, Jun 08 2023 *)

-tan(tan(tan(1))) \\ Charles R Greathouse IV, Mar 25 2014

A280092 Pierce Expansion of tan(1).

Original entry on oeis.org

1, 1, 2, 8, 12, 44, 51, 298, 934, 1041, 2267, 2668, 13150, 28929, 42652, 610672, 1630027, 2535276, 5451792, 14793658, 32179208, 38644893, 45925185, 59151658, 80924233, 118811249, 273877246, 611892649, 1688111599, 11265859421, 22104337881, 31986803107, 104253472513

Offset: 0

G. C. Greubel, Dec 25 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pierce Expansion
Wikipedia, Engel Expansion

Cf. A049471 (tan(1)).

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Tan[1] , 7!], 50]

cp's OEIS Frontend

A093178 If n is even then 1, otherwise n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A073449 Decimal expansion of cot(1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A073447 Decimal expansion of csc(1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A073448 Decimal expansion of sec(1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A009001 Expansion of e.g.f: (1+x)*cos(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A161011 Decimal expansion of tan(1/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348140 a(n) is the numerator of tan(n * arctan(1/n)).

Views

Author