A073449 -id:A073449 - cp's OEIS frontend

A049471 Decimal expansion of tan(1).

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

Offset: 1

Albert du Toit (dutwa(AT)intekom.co.za), N. J. A. Sloane

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

			1.5574077246549022305...

Harry J. Smith, Table of n, a(n) for n = 1..20000
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. A093178 (continued fraction), A009001, A073449.

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

default(realprecision, 20080); x=tan(1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b049471.txt", n, " ", d)); \\

Equals Sum_{k>=1} (-1)^(k+1) * B(2*k) * 2^(2*k) * (2^(2*k) - 1) / (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

A019987 Decimal expansion of tangent of 89 degrees.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane

Also cotangent of 1 degree. - Mohammad K. Azarian, Jan 21 2006
An algebraic integer of degree 24. - Charles R Greathouse IV, Aug 27 2017
The least k > 0 such that floor(cot(1°)*10^k) is prime is k = 39. - M. F. Hasler, May 19 2023

			57.28996...

Ivan Panchenko, Table of n, a(n) for n = 2..1000.
Art of Problem Solving, USAMO 1996, Problem 1.
Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, p. 413-414; Solution, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.
User ssaamil, I found a cool number I'd like to share, 57289961630759424687278147537112577980217. In reddit.com/r/math, May 12 2023
Index to sequences related to Olympiads.
Index entries for algebraic numbers, degree 24

Cf. A073449, A019899-A019986 (same for 1, ..., 88 degrees).

evalf(cot(Pi/180),100); # Bernard Schott, Apr 30 2022

First[RealDigits[Cot[Pi/180], 10, 100]] (* Paolo Xausa, Apr 23 2024 *)

tan(89*Pi/180) \\ Charles R Greathouse IV, Aug 27 2017

Equals (Sum_{k=1..90} 2*k*sin(2*k)) / 90, with k in degrees (link USAMO 1996). - Bernard Schott, Apr 30 2022

A largest of the 24 real-valued roots of 1 -48*x +x^24 -564*x^22 +21186*x^20 -269412*x^18 +1470447*x^16 -3923304*x^14 +5407388*x^12 -3923304*x^10 +1470447*x^8- 269412*x^6 +21186*x^4 -564*x^2 -48*x^23 +1456*x^21 -12432*x^19 +17424*x^17 +45344*x^15 -51744*x^13 -51744*x^11 +45344*x^9 +17424*x^7 -12432*x^5 +1456*x^3 =0.- R. J. Mathar, Aug 29 2025

Edited by N. J. A. Sloane, Aug 19 2008 at the suggestion of R. J. Mathar

A370189 Imaginary part of (1 + n*i)^n, where i is the imaginary unit.

Original entry on oeis.org

0, 1, 4, -18, -240, 1900, 42372, -482552, -14970816, 222612624, 8825080100, -161981127968, -7809130867824, 170561613679808, 9678967816041188, -245159013138710400, -16000787866533953280, 461102348510408544512, 34017524842099233036996, -1098983344602124698522112, -90417110945911655996319600

Offset: 0

Hugo Pfoertner, Feb 14 2024

The ratio a(n)/A121626(n) converges to c for odd n and to -1/c for even n for n -> oo with c = 0.6420926... (= cot(1) (A073449) from Moritz Firsching, Feb 14 2024). See linked plots.

Paolo Xausa, Table of n, a(n) for n = 0..350
Hugo Pfoertner, Plot of ratio a(n)/A121626(n), using Plot 2.
Hugo Pfoertner, Plot of asinh(a(n)) vs asinh(A121626(n)), using Plot 2.

Cf. A121626 (real part), A115415, A115416.

Cf. A073449.

Array[Im[(1+#*I)^#] &, 25, 0] (* Paolo Xausa, Feb 19 2024 *)

PARI
```
a370189(n) = imag((1+I*n)^n)
```

from math import comb
def A370189(n): return sum(comb(n,j)*n**j*(-1 if j-1&2 else 1) for j in range(1,n+1,2)) # Chai Wah Wu, Feb 15 2024

a(n) = Sum_{j=0..floor((n-1)/2)} binomial(n,2*j+1)*n^(2*j+1)*(-1)^j. - Chai Wah Wu, Feb 15 2024

A113815 Decimal expansion of (cotangent of 1 degree)^(1/5).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Jan 22 2006

			Equals (cot(Pi/180))^(1/5) = 2.247065367379405144530132....

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

Cf. A073449; A019987.

RealDigits[(Cot[Pi/180])^(1/5), 10, 50][[1]] (* G. C. Greubel, Jun 08 2017 *)
RealDigits[Surd[Cot[1 Degree],5],10,120][[1]] (* Harvey P. Dale, Mar 09 2019 *)

(1/tan(Pi/180))^(1/5) \\ G. C. Greubel, Jun 08 2017

A280094 Pierce Expansion of cot(1).

Original entry on oeis.org

1, 2, 3, 6, 8, 12, 13, 15, 17, 19, 1063, 1155, 2574, 2662, 3595, 3723, 4370, 21530, 28927, 32662, 73255, 92895, 5133189, 13626701, 17852908, 392678721, 715595109, 3993107840, 39941257169, 43578446054, 1686996293054, 4244526044926, 78467829696572, 111290944386765

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. A073449 (cot(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[Cot[1] , 7!], 50]

A113794 Decimal expansion of (cotangent of 1 degree)^2.

Original entry on oeis.org

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

Offset: 4

Mohammad K. Azarian, Jan 21 2006

			3282.139703... = 1/A111493.

Cf. A073449, A019987.

cot(Pi/180)^2; evalf(%) ; # R. J. Mathar, Apr 03 2011

RealDigits[Cot[1 Degree]^2,10,120][[1]] (* Harvey P. Dale, Mar 12 2023 *)

Offset changed by Mohammad K. Azarian, Dec 13 2008

A113809 Decimal expansion of (cotangent of 1 degree)^3.

Original entry on oeis.org

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

Offset: 6

Mohammad K. Azarian, Jan 22 2006

			188033.65768912...

Cf. A073449; A019987.

evalf(cot(Pi/180)^3) ; # R. J. Mathar, Oct 03 2011

RealDigits[Cot[1 Degree]^3,10,120][[1]] (* Harvey P. Dale, Nov 06 2019 *)

Offset changed by Mohammad K. Azarian, Dec 13 2008

A113810 Decimal expansion of (cotangent of 1 degree)^4.

Original entry on oeis.org

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

Offset: 8

Mohammad K. Azarian, Jan 22 2006

			0.10772441034301..*10^8.

Cf. A073449, A019987.

Offset corrected by Mohammad K. Azarian, Dec 13 2008

cp's OEIS Frontend

A049471 Decimal expansion of tan(1).

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A073447 Decimal expansion of csc(1).

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A073448 Decimal expansion of sec(1).

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A019987 Decimal expansion of tangent of 89 degrees.

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A370189 Imaginary part of (1 + n*i)^n, where i is the imaginary unit.

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A113815 Decimal expansion of (cotangent of 1 degree)^(1/5).

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A280094 Pierce Expansion of cot(1).

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