A179587 -id:A179587 - cp's OEIS frontend

A385258 Decimal expansion of the volume of a gyroelongated square bicupola with unit edge.

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

Offset: 1

Paolo Xausa, Jun 26 2025

The gyroelongated square bicupola is Johnson solid J_45.

			8.1535748336212634025260213162662727026732149...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated square bicupola.

Cf. A385259 (surface area).

Cf. A002193, A010466, A179587, A384214, A384287, A385256.

First[RealDigits[2/3*(3 + 2*# + Sqrt[2*(2 + # + Sqrt[146 + 103*#])]) & [Sqrt[2]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J45", "Volume"], 10, 100]]

Equals (2/3)*(3 + 2*sqrt(2) + sqrt(2*(2 + sqrt(2) + sqrt(146 + 103*sqrt(2))))) = (2/3)*(3 + A010466 + sqrt(2*(2 + A002193 + sqrt(146 + 103*A002193)))).

Equals the largest real root of 6561*x^8 - 104976*x^7 + 594864*x^6 - 1384128*x^5 - 552096*x^4 + 1569024*x^3 + 246528*x^2 - 119808*x + 4352.

A179593 Decimal expansion of the volume of pentagonal rotunda with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal rotunda: 20 vertices, 35 edges, and 17 faces.

			6.91776296812470206991299603070264133354087600944966144271710443099823...

Wolfram Alpha, Johnson solid 6
Index entries for algebraic numbers, degree 2.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590, A179591, A179592.

Mathematica
```
RealDigits[N[(45+17*Sqrt[5])/12,200]]
```

Digits of (45+17*sqrt(5))/12.

A179592 Decimal expansion of the circumradius of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal cupola: 15 vertices, 25 edges, and 12 faces.

			2.232950509415690049500415383249682772934080730579181647457441260...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 4.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590, A179591.

Mathematica
```
RealDigits[N[Sqrt[11+4*Sqrt[5]]/2,200]]
```

Digits of sqrt(11+4*sqrt(5))/2.

A179639 Decimal expansion of the volume of gyroelongated pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated pentagonal pyramid: 11 vertices,25 edges,and 16 faces.

			1.88019215822908780282010679244089525495689855152098881326825313369561...

Wolfram Alpha, Johnson solid 11

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637, A179638.

Mathematica
```
RealDigits[N[(25+9*Sqrt[5])/24,200]]
```

Digits of (25+9*sqrt(5))/24.

A179640 Decimal expansion of the surface area of gyroelongated pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated pentagonal pyramid: 11 vertices, 25 edges, and 16 faces.

			8.21566792897225677348693575803563097544289387179912568441637087996861...

Wolfram Alpha, Johnson solid 11

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637, A179638, A179639.

RealDigits[N[Sqrt[5/2*(70+Sqrt[5]+3*Sqrt[75+30*Sqrt[5]])]/2,200]]

Digits of sqrt(5/2*(70+sqrt(5)+3*sqrt(75+30*sqrt(5))))/2.

A179641 Decimal expansion of the volume of pentagonal dipyramid with edge length 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Pentagonal dipyramid: 7 vertices, 15 edges, and 10 faces.

			0.60300566479164914136743113906093968628671819663429381035590810378421...

Wolfram Alpha, Johnson solid 13

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449-A179452, A179552-A179554, A179587-A179593, A179637-A179640.

Mathematica
```
RealDigits[N[(5+Sqrt[5])/12,200]]
```

(sqrt(5)+5)/12 \\ Charles R Greathouse IV, Apr 25 2016

Digits of (5+sqrt(5))/12.

Offset corrected by R. J. Mathar, Aug 15 2010

A195433 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 18 2011

See A195304 for definitions and a general discussion.

			(A)=0.6147572272333906215933192480911...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A195304, A195434, A195435, A195436.

a = 1; b = 1; h = 2 a/3; k = b/3;
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195433 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (B)=(2/3)sqrt(2); -1+A179587 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100] (* (C) A195433 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195436 *)

A195436 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,1,sqrt(2) right triangle ABC.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 18 2011

See A195304 for definitions and a general discussion.

			Philo(ABC,G)=0.636258821061838308391049464710...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A195304.

a = 1; b = 1; h = 2 a/3; k = b/3;
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195433 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (B)=sqrt(8/9), -1+A179587  *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100] (* (C) A195433 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195436 *)

A378388 Decimal expansion of the surface area of a tetrakis hexahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 27 2024

The tetrakis hexahedron is the dual polyhedron of the truncated octahedron.

			11.925695879998878380848926233233473255683297917928...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Tetrakis Hexahedron.

Cf. A374359 (volume - 1), A010532 (inradius*10), A179587 (midradius + 1), A378389 (dihedral angle).
Cf. A377341 (surface area of a truncated octahedron with unit edge).
Cf. A002163, A208899.

First[RealDigits[16*Sqrt[5]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TetrakisHexahedron", "SurfaceArea"], 10, 100]]

Equals (16/3)*sqrt(5) = (16/3)*A002163 = 16*A208899.

A386459 Decimal expansion of the volume of an augmented truncated cube with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 22 2025

The augmented truncated cube is Johnson solid J_66.

			15.5424723326565069269423398624517230857049...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Augmented truncated cube.

Cf. A386460 (surface area).

Cf. A131594, A179587, A377299, A386411.

First[RealDigits[8 + 16*Sqrt[2]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J66", "Volume"], 10, 100]]

Equals 8 + 16*sqrt(2)/3 = 8 + 16*A131594.
Equals A377299 + A179587.
Equals the largest root of 9*x^2 - 144*x + 64.

cp's OEIS Frontend

A385258 Decimal expansion of the volume of a gyroelongated square bicupola with unit edge.

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A179593 Decimal expansion of the volume of pentagonal rotunda with edge length 1.

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A179592 Decimal expansion of the circumradius of pentagonal cupola with edge length 1.

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A179639 Decimal expansion of the volume of gyroelongated pentagonal pyramid with edge length 1.

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A179640 Decimal expansion of the surface area of gyroelongated pentagonal pyramid with edge length 1.

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A179641 Decimal expansion of the volume of pentagonal dipyramid with edge length 1.

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A195433 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)).

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A195436 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,1,sqrt(2) right triangle ABC.