A011002 -id:A011002 - cp's OEIS frontend

A011042 Decimal expansion of 4th root of 48.

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

Offset: 1

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 1..1000

RealDigits[Power[48, (4)^-1],10,120][[1]] (* Harvey P. Dale, Jul 26 2011 *)

sqrtn(48,4) \\ Charles R Greathouse IV, Apr 15 2014

Equals 2*A011002. [From R. J. Mathar, Feb 04 2009]

A018052 Powers of fourth root of 3 rounded to nearest integer.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 27, 36, 47, 62, 81, 107, 140, 185, 243, 320, 421, 554, 729, 959, 1263, 1662, 2187, 2878, 3788, 4985, 6561, 8635, 11364, 14956, 19683, 25904, 34092, 44868, 59049, 77713

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A011002, A018053.

Floor[# + 1/2]&/@((Power[3, (4)^-1])^Range[0, 50]) (* Vincenzo Librandi, Apr 13 2017 *)
Round[Surd[3,4]^Range[0,50]] (* Harvey P. Dale, Apr 13 2022 *)

A159824 Continued fraction for Pi^Pi (cf. A073233).

Original entry on oeis.org

36, 2, 6, 9, 2, 1, 2, 5, 1, 1, 6, 2, 1, 291, 1, 38, 50, 1, 2, 5, 4, 1, 2, 2, 1, 5, 1, 4, 13, 2, 1, 4, 3, 3, 1, 2, 25, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 1, 43, 1, 2, 7, 3, 1, 1, 1, 2, 4, 2, 1, 1, 3, 1, 3, 3, 2, 2, 16, 3, 5, 2, 1, 5, 2, 1, 10, 1, 1, 3, 1, 13, 1, 1, 3, 1, 10, 4, 1, 1, 1, 38, 1, 2, 2, 1, 1, 3

Offset: 0

Harry J. Smith, Apr 30 2009

nonn

			36.4621596072079117709908260... = 36 + 1/(2 + 1/(6 + 1/(9 + 1/(2 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A001076, A001203, A001204, A002945, A010767, A011002, A011003, A073233, A073238, A179613, A179615, A179616, A179617.

ContinuedFraction[Pi^Pi,200] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2010 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^Pi); for (n=1, 20001, write("b159824.txt", n-1, " ", x[n])); }

Edited by N. J. A. Sloane, Jul 22 2010

A179275 Decimal expansion of 2*sqrt(Pi)/3^(1/4).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jul 07 2010

Also the side length of an equilateral triangle with area Pi (A000796), the area of a unit circle.

The area of an equilateral triangle with side length s is (sqrt(3)/4)s^2 = A120011*s^2, so A120011*(this constant)^2 = A000796.

			2.693547374177196721238160475092328667088670807308015892399206645495191607305...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A002161 (sqrt(Pi)), A011002 (3^1/4), A000796 (Pi), A002194 (sqrt(3)), A120011 (sqrt(3)/4).

RealDigits[2*Sqrt[Pi]/3^(1/4), 10, 100][[1]] (* G. C. Greubel, Mar 24 2017 *)

PARI
```
2*sqrt(Pi)/3^(1/4)
```

2*sqrt(Pi)/3^(1/4) = 2*A002161/A011002.

A269430 Decimal expansion of (1 + Pi)/2.

Original entry on oeis.org

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

Offset: 1

Jani Melik, Feb 26 2016

			2.0707963267948966192313216916397514420985846996875529...

Index entries for transcendental numbers

Cf. A000796, A002193, A002194, A002163, A174968, A001622, A002161, A010767, A011002, A011003, A191502, A053004.

PARI
```
(1 + Pi)/2 \\ Altug Alkan, Apr 07 2016
```
Sage
```
N((1+pi)/2, digits=110)
```

Equals A096444 + 1 or A019669 + 1/2.

Equals Sum_{k>=0} 2^k/binomial(2*k+2,k). - Amiram Eldar, Jun 30 2020

A358186 Decimal expansion of the positive real root r of 3*x^4 - 1.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Dec 04 2022

The other roots are -r and the complex conjugate pair r*i and -r*i.

			0.75983568565159254733118775065454533539677344888731031861128865370032223317...

Cf. A011002, A011022.

RealDigits[Surd[1/3, 4], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)

r = (1/3)^(1/4) = (1/3)*27^(1/4).
Equals A011022/3.
Equals exp(-arctanh(1/2)/2). - Amiram Eldar, Jul 06 2023
Equals 1/A011002. - Jason Yuen, Jul 27 2024

A369499 Decimal expansion of exp(sqrt(3)*Pi/18)/3^(1/4).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 25 2024

			1.02803254168957677046288435785772167789241862654002...

Albert Stadler, Problem 11677, Problems and Solutions, The American Mathematical Monthly, Vol. 119, No. 10 (2012), p. 880; Dedekind eta Function Disguised, Solution to Problem 11677 by Radouan Boukharfane, ibid., Vol. 121, No. 10 (December 2014), pp. 951-952. Note that the problem appeared with typos.
Allen Stenger, Experimental Math for Math Monthly Problems, The American Mathematical Monthly, Vol. 124, No. 2 (2017), pp. 116-131; alternative link. See pp. 129-130.

Cf. A011002.

RealDigits[Exp[Sqrt[3]*Pi/18]/3^(1/4), 10, 120][[1]]

PARI
```
exp(sqrt(3)*Pi/18)/sqrtn(3, 4)
```

Equals Product_{k>=1} (1 + 2*exp(-k*Pi*sqrt(3)) * cosh(k*Pi/sqrt(3))) (Stadler, 2012).

cp's OEIS Frontend

A011042 Decimal expansion of 4th root of 48.

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A018052 Powers of fourth root of 3 rounded to nearest integer.

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A159824 Continued fraction for Pi^Pi (cf. A073233).

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A179275 Decimal expansion of 2*sqrt(Pi)/3^(1/4).

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A269430 Decimal expansion of (1 + Pi)/2.

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A358186 Decimal expansion of the positive real root r of 3*x^4 - 1.

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A369499 Decimal expansion of exp(sqrt(3)*Pi/18)/3^(1/4).

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