A010487 -id:A010487 - cp's OEIS frontend

A195289 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

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

Offset: 0

Clark Kimberling, Sep 14 2011

See A195284 for definitions and a general discussion.

			Philo(ABC,I)=0.4847823853661753483351654180...

Cf. A195284.

a = 5; b = 12; c = 13;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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) A195286 *)
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] (* (B) A195288 *)
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] (* (C) A010487 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(A,B,C,I) A195289 *)

A387321 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 27 2025

This is the dihedral angle between adjacent triangular faces at the edge where the antiprism and cupola parts of the solid meet.

Also the analogous dihedral angle in a gyroelongated square bicupola (Johnson solid J_45).

			2.6412090003740395440214510528511358326798716782548...

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

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387322, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A002193, A010487, A195696.

First[RealDigits[ArcSec[Sqrt[3]] + ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J23", "DihedralAngles"]],2], 10, 100]]

Equals arcsec(sqrt(3)) + arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = A195696 + arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A195696 + A387323.

A387322 Decimal expansion of the fourth largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 29 2025

This is the dihedral angle between a triangular face and a square face at the edge where the antiprism and cupola parts of the solid meet.

Also the analogous dihedral angle in a gyroelongated square bicupola (Johnson solid J_45).

			2.4712905456469785754732547961552537994857493330886...

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

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387321, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A003881, A002193, A010487.

First[RealDigits[Pi/4 + ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J23", "DihedralAngles"]], 4], 10, 100]]

Equals Pi/4 + arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = A003881 + arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A003881 + A387323.

A387323 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 29 2025

This is the dihedral angle between a triangular face and the octagonal face.

			1.6858923822495302658575939503353780784364569832448...

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

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387321, A387322.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A002193, A003881, A010487, A195696.

First[RealDigits[ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J23", "DihedralAngles"]], 10, 100]]

Equals arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A387321 - A195696 = A387322 - A003881.

A010130 Continued fraction for sqrt(32).

Original entry on oeis.org

5, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10

Offset: 0

N. J. A. Sloane

			5.65685424949238019520675489... = 5 + 1/(1 + 1/(1 + 1/(1 + 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 (0,0,0,1).

Cf. A010487 (decimal expansion).

Cf. A041052/A041053 (convergents), A248259 (Egyptian fraction).

ContinuedFraction[Sqrt[32],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{5},100,{10,1,1,1}] (* Harvey P. Dale, Aug 20 2014 *)

{ allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(32)); for (n=0, 20000, write("b010130.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2) = 1, a(2^e) = 10 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 9/4^s). (End)
From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: (5 + x + x^2 + x^3 + 5*x^4)/((1 - x)*(1 + x + x^2 + x^3)).
a(n) = a(n-4), n > 4. (End)

A041052 Numerators of continued fraction convergents to sqrt(32).

Original entry on oeis.org

5, 6, 11, 17, 181, 198, 379, 577, 6149, 6726, 12875, 19601, 208885, 228486, 437371, 665857, 7095941, 7761798, 14857739, 22619537, 241053109, 263672646, 504725755, 768398401, 8188709765, 8957108166, 17145817931, 26102926097, 278175078901, 304278004998

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,34,0,0,0,-1).

Cf. A010487, A041053.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[32], n]]], {n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Numerator[Convergents[Sqrt[32], 30]] (* Vincenzo Librandi, Oct 23 2013 *)

G.f.: (5+6*x+11*x^2+17*x^3+11*x^4-6*x^5+5*x^6-x^7)/(1-34*x^4+x^8). - Colin Barker, Jan 03 2012

A365165 Length of the perimeter of the regular 9-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 24 2023

			6.1563625798620371947937930642806724537...

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

Cf. A019845 (5-gon), A010487 (4-gon), A365163 (7-gon), A365164 (8-gon), A272488 (edge length).

First[RealDigits[18*Sin[Pi/9], 10, 100]] (* Paolo Xausa, Mar 19 2024 *)

Equals 9*A272488.

A384287 Decimal expansion of the volume of a square orthobicupola with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 05 2025

The square orthobicupola is Johnson solid J_28.

Also the volume of a square gyrobicupola (Johnson solid J_29) with unit edge.

			3.885618083164126731735584965612930771426229167169...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Square gyrobicupola.
Wikipedia, Square orthobicupola.

Cf. A010469 (surface area - 10).

Cf. A002193, A010487, A384624.

First[RealDigits[2 + Sqrt[32]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J28", "Volume"], 10, 100]]

Equals 2 + (4/3)*sqrt(2) = 2 + (4/3)*A002193 = 2 + A010487/3.

Equals the largest root of 9*x^2 - 36*x + 4.

A365163 Length of the perimeter of the regular heptagon with unit circumradius.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 24 2023

			6.0743723476458136866607566598...

Cf. A019845 (5-gon), A010487 (4-gon), A365164 (8-gon), A365165 (9-gon), A272487 (edge length), A104957 (area).

f[R_, s_] := 2*R*s*Sin[Pi/s]; a[n_] := RealDigits[f[1, 7], 10, n][[1]]; a[92] (* Robert P. P. McKone, Aug 24 2023 *)

Equals 7*A272487.

A365164 Length of the perimeter of the regular octagon with unit circumradius.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 24 2023

			6.122934917841436347655359744486...

Cf. A019845 (5-gon), A010487 (4-gon), A365163 (7-gon), A365165 (9-gon), A101464 (edge length), A010466 (area).

f[R_, s_] := 2*R*s*Sin[Pi/s]; a[n_] := RealDigits[f[1, 8], 10, n][[1]]; a[109] (* Robert P. P. McKone, Aug 24 2023 *)

Equals 8*A101464.

cp's OEIS Frontend

A195289 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

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A387321 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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A387322 Decimal expansion of the fourth largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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A387323 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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A010130 Continued fraction for sqrt(32).

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A041052 Numerators of continued fraction convergents to sqrt(32).

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A365165 Length of the perimeter of the regular 9-gon with unit circumradius.

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A384287 Decimal expansion of the volume of a square orthobicupola with unit edge.

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