A002163 -id:A002163 - cp's OEIS frontend

A384285 Decimal expansion of the volume of a gyroelongated pentagonal rotunda with unit edge.

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

Offset: 2

Paolo Xausa, May 29 2025

The gyroelongated pentagonal rotunda is Johnson solid J_25.

			13.667050843671696932123530899233286565400264...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated pentagonal rotunda.

Cf. A384286 (surface area).

Cf. A002163, A179590, A179639, A179641, A384138, A384140, A384144, A384213, A384283.

First[RealDigits[(45 + 17*# + 10*Sqrt[2*(Sqrt[650 + 290*#] - # - 1)])/12 & [Sqrt[5]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J25", "Volume"], 10, 100]]

Equals (45 + 17*sqrt(5) + 10*sqrt(2*(sqrt(650 + 290*sqrt(5)) - sqrt(5) - 1)))/12 = (45 + 17*A002163 + 10*sqrt(2*(sqrt(650 + 290*A002163) - A002163 - 1)))/12.

Equals the largest real root of 1679616*x^8 - 50388480*x^7 + 603262080*x^6 - 3520972800*x^5 + 5215460400*x^4 + 4128624000*x^3 - 8894943000*x^2 + 3881385000*x - 424924375.

A384286 Decimal expansion of the surface area of a gyroelongated pentagonal rotunda with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, May 30 2025

The gyroelongated pentagonal rotunda is Johnson solid J_25.

			31.00745430323851474443564586571797490853203978248...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated pentagonal rotunda.

Cf. A384285 (volume).

Cf. A002163, A002194, A179553, A179591, A179640, A384141, A384284.

First[RealDigits[(15*Sqrt[3] + Sqrt[650 + 290*Sqrt[5]])/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J25", "SurfaceArea"], 10, 100]]

Equals (15*sqrt(3) + sqrt(650 + 290*sqrt(5)))/2 = (15*A002194 + sqrt(650 + 290*A002163))/2.

Equals the largest root of 256*x^8 - 339200*x^6 + 98924000*x^4 - 9264250000*x^2 + 176295015625.

A384871 Decimal expansion of the volume of a pentagonal orthocupolarotunda with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 11 2025

The pentagonal orthocupolarotunda is Johnson solid J_32.

Also the volume of a pentagonal gyrocupolarotunda (Johnson solid J_33) with unit edge.

			9.2418082864578952008524451431901588238346215825240...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Pentagonal gyrocupolarotunda.
Wikipedia, Pentagonal orthocupolarotunda.

Cf. A384872 (surface area).

Cf. A002163, A384144, A384283, A384285, A384624.

First[RealDigits[5*(11 + 5*Sqrt[5])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J32", "Volume"], 10, 100]]

Equals (5/12)*(11 + 5*sqrt(5)) = (5/12)*(11 + 5*A002163).

Equals the largest root of 36*x^2 - 330*x - 25.

A385802 Decimal expansion of the volume of a parabiaugmented dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 09 2025

The parabiaugmented dodecahedron is Johnson solid J_59.

Also the volume of a metabiaugmented dodecahedron (Johnson solid J_60) with unit edge.

			8.266124625416281110083485059340673098307800325954...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Parabiaugmented dodecahedron.

Cf. A385803 (surface area).

Cf. A002163, A102769, A179552, A385695, A385804.

First[RealDigits[(25 + 11*Sqrt[5])/6, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J59", "Volume"], 10, 100]]

Equals (25 + 11*sqrt(5))/6 = (25 + 11*A002163)/6.
Equals A102769 + 2*A179552.
Equals the largest root of 9*x^2 - 75*x + 5.

A068446 Factorial expansion of sqrt(5) = Sum_{n>0} a(n)/n!.

Original entry on oeis.org

2, 0, 1, 1, 3, 1, 6, 6, 2, 3, 5, 2, 12, 1, 7, 1, 3, 10, 12, 19, 10, 18, 21, 6, 3, 10, 10, 26, 18, 0, 26, 30, 5, 21, 21, 5, 28, 34, 22, 9, 28, 32, 0, 9, 19, 20, 8, 9, 16, 43, 28, 22, 4, 40, 54, 17, 51, 55, 31, 18, 52, 37, 55, 0, 45, 61, 16, 41, 62, 53, 20, 31, 49, 63, 62, 20, 69, 1, 64

Offset: 1

Benoit Cloitre, Mar 10 2002

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system: Fractional values
Index to sequences related to factorial base representation of noninteger constants.

Cf. A002163 (decimal expansion of sqrt(5)).

[Floor(Sqrt(5))] cat [Floor(Factorial(n)*Sqrt(5)) - n*Floor(Factorial((n-1))*Sqrt(5)) : n in [2..30]]; // G. C. Greubel, Mar 21 2018

Table[If[n==1, Floor[Sqrt[5]],Floor[n!*Sqrt[5]]-n*Floor[(n-1)!*Sqrt[5] ]], {n,1,50}] (* G. C. Greubel, Mar 21 2018 *)

for(n=1,30, print1(if(n==1, floor(sqrt(5)), floor(n!*sqrt(5)) - n*floor((n-1)!*sqrt(5))), ", ")) \\ G. C. Greubel, Mar 21 2018

A068446_vec(N=90,c=sqrt(precision(5.,N*log(N/2.4)\/2.3)))=vector(N,n, if(n>1,c=c%1*n,c)\1) \\ M. F. Hasler, Nov 27 2018

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a regular dodecahedron to the volume of the circumscribed sphere (which has circumradius a*(sqrt(3) + sqrt(15))/4 = a*(A002194 + A010472)/4, where a is the dodecahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165954. A063723 shows the order of these by size.

			0.6649088942053266431144284467086337161648765805556919381057592605722964718...

Eric Weisstein's World of Mathematics, Dodecahedron.
Index entries for transcendental numbers.

Cf. A000796, A002194, A010472, A165922, A049541, A165952, A165954, A063723, A002163, A020760, A010527.

RealDigits[(5*Sqrt[3]+Sqrt[15])/(6*Pi),10,120][[1]] (* Harvey P. Dale, Feb 16 2018 *)

PARI
```
(5*sqrt(3)+sqrt(15))/(6*Pi)
```

Equals (5*A002194 + A010472)/(6*A000796).
Equals (5*A002194 + A010472)*A049541/6.
Equals (10*A010527 + A010472)*A049541/6.
Equals (5 + sqrt(5))/(2*Pi*sqrt(3)).
Equals (5 + A002163)*A049541*A020760/2.

A165954 Decimal expansion of sqrt(10 + 2sqrt(5))/(2Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 04 2009

The ratio of the volume of a regular icosahedron to the volume of the circumscribed sphere (with circumradius a*sqrt(10 + 2*sqrt(5))/4 = a*A019881, where a is the icosahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165953. A063723 shows the order of these by size.

			0.6054613829125255833862652051280444903008454088014288933200935000838295683...

Eric Weisstein's World of Mathematics, Icosahedron.
Index entries for transcendental numbers.

Cf. A000796, A002163, A165922, A049541, A165952, A165953, A063723, A019881, A019694, A060294.

RealDigits[Sqrt[10+2Sqrt[5]]/(2Pi),10,120][[1]] (* Harvey P. Dale, Aug 27 2013 *)

PARI
```
sqrt(10+2*sqrt(5))/(2*Pi)
```

sqrt(10 + 2*sqrt(5))/(2*Pi) = sqrt(10 + 2*A002163)/(2*A000796) = 2*sin(2*Pi/5)/Pi = 2*sin(A019694)/A000796 = 2*sin(72 deg)/Pi = 2*A019881/A000796 = 2*A019881*A049541 = (2/Pi)*sin(72 deg) = A060294*A019881.

A176322 Decimal expansion of sqrt(1365).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of sqrt(1365) is (repeat 1, 17, 2, 17, 1, 72) preceded by 36.

			36.94590640382233199163...

G. C. Greubel, Table of n, a(n) for n = 2..1001

Cf. A002194 (sqrt(3)), A002163 (sqrt(5)), A010465 (sqrt(7)), A010470 (sqrt(13)).

Cf. A176321 ((35+sqrt(1365))/14).

SetDefaultRealField(RealField(120)); Sqrt(1365); // G. C. Greubel, Nov 26 2019

evalf( sqrt(1365), 120); # G. C. Greubel, Nov 26 2019

RealDigits[Sqrt[1365],10,120][[1]] (* Harvey P. Dale, Oct 09 2017 *)

default(realprecision, 120); sqrt(1365) \\ G. C. Greubel, Nov 26 2019

numerical_approx(sqrt(1365), digits=120) # G. C. Greubel, Nov 26 2019

Equals sqrt(3)*sqrt(5)*sqrt(7)*sqrt(13).

A200991 Decimal expansion of square root of 221/25.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Dec 06 2011

This is the third Lagrange number, corresponding to the third Markov number (5). With multiples of the golden ration and sqrt(2) excluded from consideration, the Hurwitz irrational number theorem uses this Lagrange number to obtain very good rational approximations for irrational numbers.

Continued fraction is 2 followed by 1, 36, 3, 148, 3, 36, 1, 4 repeated.

			2.9732137494637011045224016...

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 187

Eric Weisstein's World of Mathematics, Lagrange Number.

Cf. A002163 (the first Lagrange number), A010466 (the second Lagrange number).

Mathematica
```
RealDigits[Sqrt[221/25], 10, 100][[1]]
```

sqrt(221)/5 \\ Charles R Greathouse IV, Dec 06 2011

With m = 5 being a Markov number (A002559), L = sqrt(9 - 4/m^2).

A204188 Decimal expansion of sqrt(5)/4.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Jan 14 2012

Equals Product_{n>=1} (1 - 1/A000032(2^n)).
Essentially the same as A019863 and A019827. - R. J. Mathar, Jan 16 2012
The value is the distance of the W point of the Wigner-Seitz cell of the body-centered cubic lattice (that is the Brioullin zone of the face-centered cubic lattice) to its four nearest neighbors. Let the points of the simple cubic lattice be at (1,0,0), (0,1,0), (1,0,0) etc and the point in the cube center at (1/2, 1/2, 1/2). Then W is at (0, 1/4, 1/2) [or any of the 24 symmetry related positions like (0, 3/4, 1/2), (0, 1/2, 1/4) etc.], and the four lattice points closest to W are at (-1/2, 1/2, 1/2), (0,0,0), (1/2, 1/2, 1/2) and (0,0,1). - R. J. Mathar, Aug 19 2013

			0.5590169943749474241022934171828190588601545899028814310677243113526302...

J. Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
Y. Tachiya, Transcendence of certain infinite products, J. Number Theory 125 (2007), 182-200.
Wikipedia, Brillouin zone

Cf. A001622, A002163.

Maple
```
evalf(sqrt(5)/4);
```

RealDigits[Sqrt[5]/4, 10, 100][[1]] (* Amiram Eldar, Dec 04 2018 *)

sqrt(5)/4 \\ Charles R Greathouse IV, Apr 21 2016

Equals sqrt(5)/4 = (-1 + 2*phi)/4, with the golden section phi from A001622.

Equals 5*A020837.

cp's OEIS Frontend

A384285 Decimal expansion of the volume of a gyroelongated pentagonal rotunda with unit edge.

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A384286 Decimal expansion of the surface area of a gyroelongated pentagonal rotunda with unit edge.

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A384871 Decimal expansion of the volume of a pentagonal orthocupolarotunda with unit edge.

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A385802 Decimal expansion of the volume of a parabiaugmented dodecahedron with unit edge.

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A068446 Factorial expansion of sqrt(5) = Sum_{n>0} a(n)/n!.

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A165953 Decimal expansion of (5*sqrt(3) + sqrt(15))/(6*Pi).

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A165954 Decimal expansion of sqrt(10 + 2*sqrt(5))/(2*Pi).

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A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

A165954 Decimal expansion of sqrt(10 + 2sqrt(5))/(2Pi).