A010482 -id:A010482 - cp's OEIS frontend

A248254 Egyptian fraction representation of sqrt(27) (A010482) using a greedy function.

5, 6, 34, 13516, 202119099, 64783216365098195, 22100984125756663557825370106132649, 666714143657173655990633057343413567220367208291412102910376204532308

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 27]]

A178809 Decimal expansion of the area of the regular 12-gon (dodecagon) of edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

Surface area of a regular hexagonal prism with unit side length and height. - Wesley Ivan Hurt, May 04 2021

			11.196152422706631880582339024517617100828415761431141884167420938355...

G. C. Greubel, Table of n, a(n) for n = 2..10000
Index entries for algebraic numbers, degree 2

Cf. A010482, A090488, A104956, A120011, A176532.

Mathematica
```
RealDigits[N[6+3*Sqrt[3],150]][[1]]
```

sqrt(27)+6 \\ Charles R Greathouse IV, Apr 25 2016

Equals 6+3*sqrt(3).

Equals 1 + A176532 = 6 + A010482. - R. J. Mathar, Jun 25 2010

Offset corrected and keyword:cons inserted by R. J. Mathar, Jun 25 2010

A131594 Decimal expansion of sqrt(2)/3, the volume of a regular octahedron with edge length 1.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Aug 30 2007

Volume of a regular octahedron: V = ((sqrt(2))/3)* a^3, where 'a' is the edge.

			0.471404520791031682933896...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A020829 (regular tetrahedron volume), A102208 (regular icosahedron volume), A102769 (regular dodecahedron volume).

Cf. A179587.

RealDigits[Sqrt[2]/3,10,120][[1]] (* Harvey P. Dale, May 27 2012 *)

PARI
```
sqrt(2)/3 \\ G. C. Greubel, Jul 06 2017
```

Equals A002193/3 = A010464/A010482. - R. J. Mathar, Dec 11 2009

More digits from R. J. Mathar, Dec 11 2009

A020784 Decimal expansion of 1/sqrt(27).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This is the minimum ripple factor for a third-order Chebyshev filter for which the generalized reflectionless topology needs no negative elements. - Matthew A. Morgan, Oct 18 2017

			0.1924500897298752548363829268339858185492005837567089586728674....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.16, pp. 495, 527.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010482, A020760, A021031, A073010, A212886.

1/Sqrt(27); // G. C. Greubel, Feb 15 2018

RealDigits[1/Sqrt[27],10,200][[1]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)
realDigitsRecip[Sqrt[27]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jul 05 2025 *)

1/sqrt(27) \\ G. C. Greubel, Feb 15 2018

Equals Sum_{k>=0} binomial(2*k,k) * k/16^k. - Amiram Eldar, Aug 02 2020
Equals sqrt(3)/9. - Stefano Spezia, Dec 24 2024
Equals 1/A010482 = A020760/3 = sqrt(A021031) = A073010/Pi = A212886/2. - Hugo Pfoertner, Dec 24 2024

A040021 Continued fraction for sqrt(27).

Original entry on oeis.org

5, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5

Offset: 0

N. J. A. Sloane

			5.1961524227066318805823390... = 5 + 1/(5 + 1/(10 + 1/(5 + 1/(10 + ...)))). - _Harry J. Smith_, Jun 04 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010482 (Decimal expansion), A010721.

ContinuedFraction(Sqrt(27)); // G. C. Greubel, Feb 16 2018

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[27],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{5},120,{10,5}] (* Harvey P. Dale, Jul 19 2015 *)

{ allocatemem(932245000); default(realprecision, 35000); x=contfrac(sqrt(27)); for (n=0, 20000, write("b040021.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

G.f.: 5*(1 + x + x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A383852 Decimal expansion of the volume of an elongated triangular pyramid with unit edge.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, May 19 2025

The elongated triangular pyramid is Johnson solid J_7.

			0.55086383208997724411533564572727626494984063640067...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Elongated triangular pyramid.

Cf. A165663 (surface area).

Cf. A002193, A010482.

First[RealDigits[(Sqrt[2] + Sqrt[27])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J7", "Volume"], 10, 100]]

Equals (sqrt(2) + 3*sqrt(3))/12 = (A002193 + A010482)/12.

Minimal polynomial: 20736*x^4 - 8352*x^2 + 625. - Stefano Spezia, May 19 2025

A041043 Denominators of continued fraction convergents to sqrt(27).

Original entry on oeis.org

1, 5, 51, 260, 2651, 13515, 137801, 702520, 7163001, 36517525, 372338251, 1898208780, 19354426051, 98670339035, 1006057816401, 5128959421040, 52295652026801, 266607219555045, 2718367847577251, 13858446457441300

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,52,0,-1).

Cf. A010482, A041042.

Denominator[Convergents[Sqrt[27],50]] (* Harvey P. Dale, Apr 22 2012 *)
CoefficientList[Series[- (x^2 - 5 x - 1)/(x^4 - 52 x^2 + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 22 2013 *)
a0[n_] := (9+5*Sqrt[3]+(9-5*Sqrt[3])*(26+15*Sqrt[3])^(2*n))/(18*(26+15*Sqrt[3])^n) // Simplify
a1[n_] := (-1+(26+15*Sqrt[3])^(2*n))/(6*Sqrt[3]*(26+15*Sqrt[3])^n) // FullSimplify
Flatten[MapIndexed[{a0[#],a1[#]}&,Range[10]]] (* Gerry Martens, Jul 10 2015 *)

a(n) = 52*a(n-2)-a(n-4). G.f.: -(x^2-5*x-1)/(x^4-52*x^2+1). - Colin Barker, Jul 15 2012
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)]:
a0(n) = ((9+5*sqrt(3))/(26+15*sqrt(3))^n+(9-5*sqrt(3))*(26+15*sqrt(3))^n)/18.
a1(n) = (-1/(26+15*sqrt(3))^n+(26+15*sqrt(3))^n)/(6*sqrt(3)). (End)

A374607 a(n) is the numerator of (1134n^3 + 2097n^2 + 1188n + 193)/(10368n^4 + 20736n^3 + 14112n^2 + 3744*n + 320).

Original entry on oeis.org

193, 1153, 20029, 832, 111073, 50077, 327757, 7816, 724513, 251857, 1355773, 55511, 2275969, 715357, 3539533, 134909, 5200897, 1549441, 7314493, 133717, 9934753, 2862973, 13116109, 233347, 16912993, 4764817, 21379837, 746297, 26571073, 7363837, 32541133, 1119851

Offset: 0

Paolo Xausa, Jul 13 2024

See Bailey and Crandall (2001), section 5 (pp. 183-185) for a derivation of this rational polynomial.

Denominators are given by A374608.

Paolo Xausa, Table of n, a(n) for n = 0..10000
David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, pp. 175-190 (preprint draft).

Cf. A000796, A010482, A089357, A374334, A374580, A374608 (denominators).

A374607[n_] := Numerator[(1134*n^3 + 2097*n^2 + 1188*n + 193)/(10368*n^4 + 20736*n^3 + 14112*n^2 + 3744*n + 320)];
Array[A374607, 50, 0]

from math import gcd
def A374607(n): return (p:=n*(n*(1134*n + 2097) + 1188) + 193)//gcd(p,n*(n*(n*(324*n + 648) + 441) + 117) + 10<<5) # Chai Wah Wu, Jul 14 2024

sqrt(27)*(Sum_{n >= 0} (1/64^n)*a(n)/A374608(n)) = A000796. See Bailey and Crandall (2001), p. 185.

A374608 a(n) is the denominator of (1134n^3 + 2097n^2 + 1188n + 193)/(10368n^4 + 20736n^3 + 14112n^2 + 3744*n + 320).

Original entry on oeis.org

320, 12320, 396032, 24035, 4222400, 2360960, 18446720, 511313, 54017600, 21079520, 125864960, 5660830, 252900032, 86027840, 458015360, 18690490, 768084800, 242991008, 1213963520, 23415035, 1830488000, 553679360, 2656476032, 49394345, 3734726720, 1095728480, 5112020480

Offset: 0

Paolo Xausa, Jul 13 2024

See Bailey and Crandall (2001), section 5 (pp. 183-185) for a derivation of this rational polynomial.

Numerators are given by A374607.

Paolo Xausa, Table of n, a(n) for n = 0..10000
David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, pp. 175-190 (preprint draft).

Cf. A000796, A010482, A089357, A374335, A374581, A374607 (numerators).

A374608[n_] := Denominator[(1134*n^3 + 2097*n^2 + 1188*n + 193)/(10368*n^4 + 20736*n^3 + 14112*n^2 + 3744*n + 320)];
Array[A374608, 50, 0]

from math import gcd
def A374608(n): return (q:=n*(n*(n*(324*n + 648) + 441) + 117) + 10<<5)//gcd(n*(n*(1134*n + 2097) + 1188) + 193,q) # Chai Wah Wu, Jul 14 2024

sqrt(27)*(Sum_{n >= 0} (1/64^n)*A374607(n)/a(n)) = A000796. See Bailey and Crandall (2001), p. 185.

A384139 Decimal expansion of the volume of an elongated triangular bipyramid with unit edges.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, May 20 2025

The elongated triangular bipyramid is Johnson solid J_14.

Also the volume of an augmented triangular prism (Johnson solid J_49) with unit edges. - Paolo Xausa, Aug 04 2025

			0.66871496228773516484880970607808443816397995934875...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Augmented triangular prism.
Wikipedia, Elongated triangular bipyramid.

Cf. A165663 (surface area - 2).

Cf. A010482, A010482, A383852, A384138, A384140.

First[RealDigits[(Sqrt[8] + Sqrt[27])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J14", "Volume"], 10, 100]]

Equals (2*sqrt(2) + 3*sqrt(3))/12 = (A010466 + A010482)/12.

Equals the largest root of 20736*x^4 - 10080*x^2 + 361.

cp's OEIS Frontend

A248254 Egyptian fraction representation of sqrt(27) (A010482) using a greedy function.

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A178809 Decimal expansion of the area of the regular 12-gon (dodecagon) of edge length 1.

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A131594 Decimal expansion of sqrt(2)/3, the volume of a regular octahedron with edge length 1.

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A020784 Decimal expansion of 1/sqrt(27).

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Magma

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A040021 Continued fraction for sqrt(27).

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Maple

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A383852 Decimal expansion of the volume of an elongated triangular pyramid with unit edge.

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A041043 Denominators of continued fraction convergents to sqrt(27).

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A374607 a(n) is the numerator of (1134n^3 + 2097n^2 + 1188n + 193)/(10368n^4 + 20736n^3 + 14112n^2 + 3744*n + 320).

A374608 a(n) is the denominator of (1134n^3 + 2097n^2 + 1188n + 193)/(10368n^4 + 20736n^3 + 14112n^2 + 3744*n + 320).