A179587 -id:A179587 - cp's OEIS frontend

A195304 Decimal expansion of shortest length of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5).

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

Offset: 1

Clark Kimberling, Sep 18 2011

The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P.  Philo lines are not generally Euclidean-constructible.
...
Suppose that P lies inside a triangle ABC.  Let (A) denote the shortest length of segment from AB through P to AC, and likewise for (B) and (C).  The Philo sum for ABC and P is here introduced as s=(A)+(B)+(C), and the Philo number for ABC and P, as s/(a+b+c), denoted by Philo(ABC,P).
...
Listed below are examples for which P=G (the centroid); in this list, r'n means sqrt(n) and t=(1+sqrt(5))/2 (the golden ratio).
a....b...c........(A).......(B)........(C)...Philo(ABC,G)
3....4....5......A195304...A195305....A105306...A195411
5....12...13.....A195412...A195413....A195414...A195424
7....24...25.....A195425...A195426....A195427...A195428
8....15...17.....A195429...A195430....A195431...A195432
1....1....r'2....A195433..-1+A179587..A195433...A195436
1....2....r'5....A195434...A195435....A195444...A195445
1....3....r'10...A195446...A195447....A195448...A195449
2....3....r'13...A195450...A195451....A195452...A195453
r'2..r'3..r'5....A195454...A195455....A195456...A195457
1....r'2..r'3....A195471...A195472....A195473...A195474
1....r'3..2......A195475...A195476....A195477...A195478
2....r'5..3......A195479...A195480....A195481...A195482
r'2..r'5..r'7....A195483...A195484....A195485...A195486
r'7..3....4......A195487...A195488....A195489...A195490
1....r't..t......A195491...A195492....A195493...A195494
t-1..t....r'3....A195495...A195496....A195497...A195498
A similar list for P=incenter is given at A195284.

			1.89630056630920201475386720365481991708010328....

Index entries for algebraic numbers, degree 6

Cf. A195305, A195306, A195307; A195284 (P=incenter).

a = 3; b = 4; 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) A195304 *)
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) A195305 *)
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) A195306 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]   (* Philo(ABC,G) A195411 *)

polrootsreal(2025*x^6 + 21429*x^4 + 4939*x^2 - 389017)[2] \\ Charles R Greathouse IV, Feb 03 2025

A179590 Decimal expansion of the volume of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

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

			2.32404531833319313093944911248751749029374557307435048284726483027368...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 2

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

Mathematica
```
RealDigits[N[(5+4*Sqrt[5])/6,200]]
```

Digits of (5+4*sqrt(5))/6.

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

A179591 Decimal expansion of the surface area of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

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

			16.5797497529881970460940463443632246181026360961176551818747440...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 8.

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

RealDigits[N[(20+Sqrt[10*(80+31*Sqrt[5]+Sqrt[2175+930*Sqrt[5]])])/4,200]]

Digits of (20+sqrt(10*(80+31*sqrt(5)+sqrt(2175+930*sqrt(5)))))/4.

A179588 Decimal expansion of the surface area of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.

			11.56047793231506739113082378992526852408214900456427677440...

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

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

Mathematica
```
RealDigits[N[7+2*Sqrt[2]+Sqrt[3],200]]
```

Digits of 7 + 2*sqrt(2) + sqrt(3).

A179637 Decimal expansion of the surface area of pentagonal rotunda with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

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

			22.3472002653941282767984141581886130738180135134316226129799763167102...

Wolfram Alpha, Johnson solid 6

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

RealDigits[N[Sqrt[5*(145+58*Sqrt[5]+2*Sqrt[30*(65+29*Sqrt[5])])]/2,200]]

Digits of sqrt(5*(145+58*sqrt(5)+2*sqrt(30*(65+29*sqrt(5)))))/2.

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

A224268 Decimal expansion of Product_{n>=1} (1 - 1/(4n+1)^2).

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Apr 02 2013

			0.9270373386506859592169251735976300231087994117608834527929640225280...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Notes on the constants A096427 and A224268.

Cf. A064853, A085565, A093341, A327996.

Cf. product(1-1/(4n+r)^2, n>=1): A096427 (r=-1), A112628 (r=0), A179587-1 (r=2).

RealDigits[N[Product[1 - 1/(4 n + 1)^2, {n, 1, Infinity}], 90]][[1]] (* or, by the formula: *) RealDigits[Gamma[1/4]^2/(8 Sqrt[Pi]), 10, 90][[1]]

prodnumrat(1 - 1/(4*n+1)^2, 1) \\ Charles R Greathouse IV, Feb 07 2025

Equals Gamma(1/4)^2/(8*sqrt(Pi)) = L/(4*sqrt(2)), where L is the Lemniscate constant (A064853).
From Peter Bala, Feb 26 2019: (Start)
C = (Pi/4)*( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.
C = (-1)^m*2^(2*m+1)/Catalan(m) * Product_{n >= 1} ( 1 - (4*m + 3)^2/(4*n + 1)^2 ), for m = 0,1,2,....
C = Integral_{x = 0..1} 1/sqrt(1 + x^4) dx.
C = (1/sqrt(2))*Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.
C = (3/2)*Integral_{x = 0..1} sqrt(1 + x^4) dx - sqrt(2)/2.
C = (1/8)*Integral_{x = 0..1} 1/(x - x^2)^(3/4) dx.
C = Sum_{n >= 0} binomial(-1/2,n)/(4*n + 1) = Sum_{n >= 0} binomial(2*n,n)/4^n * 1/(4*n + 1).
C = (1/2)*Sum_{n >= 0} (-1)^n*binomial(-3/4,n)/(4*n + 1).
Continued fraction: 1 - 1/(5 + 20/(1 + 30/(3 + ... + (4*n)*(4*n + 1)/(1 + (4*n + 1)*(4*n + 2)/(3 + ... ))))).
C = A085565/sqrt(2). C = Pi/(4*A096427). (End)
Equals A093341/2 = A327996^2. - Hugo Pfoertner, Oct 31 2024

A179589 Decimal expansion of the circumradius of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.

			1.398966325965906702031540539431998764673522563866238879930...

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

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

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

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

A179638 Decimal expansion of the volume of gyroelongated square pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated square pyramid: 9 vertices, 20 edges, and 13 faces.

			1.19270224223223255906019842837725158155262551828862015707793142188822...

Wolfram Alpha, Johnson solid 10

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

RealDigits[N[(Sqrt[2]+2*Sqrt[4+3*Sqrt[2]])/6,200]]

Digits of (sqrt(2)+2 sqrt(4+3 sqrt(2)))/6.

A384214 Decimal expansion of the volume of a gyroelongated square cupola with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, May 23 2025

The gyroelongated square cupola is Johnson solid J_23.

			6.21076579203920003665822883345980731696010032091...

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

Cf. A384215 (surface area).

Cf. A002193, A010466, A179587, A179638, A384142.

First[RealDigits[1 + Sqrt[8]/3 + 2/3*Sqrt[4 + Sqrt[8] + 2*Sqrt[146 + 103*Sqrt[2]]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J23", "Volume"], 10, 100]]

Equals 1 + (2/3)*sqrt(2) + (2/3)*sqrt(4 + 2*sqrt(2) + 2*sqrt(146 + 103*sqrt(2))) = 1 + A010466/3 + (2/3)*sqrt(4 + A010466 + 2*sqrt(146 + 103*A002193)).

Equals the largest real root of 6561*x^8 - 52488*x^7 + 113724*x^6 - 9720*x^5 - 1616922*x^4 + 396360*x^3 + 1537020*x^2 - 178632*x - 3391.

cp's OEIS Frontend

A195304 Decimal expansion of shortest length of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5).

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A179590 Decimal expansion of the volume of pentagonal cupola with 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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A179591 Decimal expansion of the surface area of pentagonal cupola with edge length 1.

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A179588 Decimal expansion of the surface area of square cupola with edge length 1.

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A179637 Decimal expansion of the surface area of pentagonal rotunda with edge length 1.

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Mathematica

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A224268 Decimal expansion of Product_{n>=1} (1 - 1/(4n+1)^2).

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