A010487 -id:A010487 - cp's OEIS frontend

A195284 Decimal expansion of shortest length of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5); i.e., decimal expansion of 2*sqrt(10)/3.

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

Offset: 1

Clark Kimberling, Sep 14 2011

Apart from the first digit, the same as A176219 (decimal expansion of 2+2*sqrt(10)/3).
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 number for ABC and P is here introduced as the normalized sum ((A)+(B)+(C))/(a+b+c), denoted by Philo(ABC,P).
...
Listed below are examples for which P=incenter (the center, I, of the circle inscribed in ABC, the intersection of the angle bisectors of ABC); in this list, r'x means sqrt(x), and t=(1+sqrt(5))/2 (the golden ratio).
a....b....c.......(A).......(B).......(C)....Philo(ABC,I)
3....4....5.....A195284...A002163...A010466...A195285
5....12...13....A195286...A195288...A010487...A195289
7....24...25....A195290...A010524...15/2......A195292
8....15...17....A195293...A195296...A010524...A195297
28...45...53....A195298...A195299...A010466...A195300
1....1....r'2...A195301...A195301...A163960...A195303
1....2....r'5...A195340...A195341...A195342...A195343
1....3....r'10..A195344...A195345...A195346...A195347
2....3....r'13..A195355...A195356...A195357...A195358
2....5....r'29..A195359...A195360...A195361...A195362
r'2..r'3..r'5...A195365...A195366...A195367...A195368
1....r'2..r'3...A195369...A195370...A195371...A195372
1....r'3..2.....A195348...A093821...A120683...A195380
2....r'5..3.....A195381...A195383...A195384...A195385
r'2..r'5..r'7...A195386...A195387...A195388...A195389
r'3..r'5..r'8...A195395...A195396...A195397...A195398
r'7..3....4.....A195399...A195400...A195401...A195402
1....r't..t.....A195403...A195404...A195405...A195406
t-1..t....r'3...A195407...A195408...A195409...A195410
...
In the special case that P is the incenter, I, each Philo line, being perpendicular to an angle bisector, is constructible, and (A),(B),(C) can be evaluated exactly.
For the 3,4,5 right triangle, (A)=(2/3)*sqrt(10), (B)=sqrt(5), (C)=sqrt(8), so that Philo(ABC,I)=((2/3)sqrt(10)+sqrt(5)+sqrt(8))/12, approximately 0.59772335.
...
More generally, for arbitrary right triangle (a,b,c) with a<=b(A)=f*sqrt(a^2+(b+c)^2)/(b+c),
(B)=f*sqrt(b^2+(c+a)^2)/(c+a),
(C)=f*sqrt(2).
It appears that I is the only triangle center P for which simple formulas for (A), (B), (C) are available. For P=centroid, see A195304.

			2.10818510677891955466592902962...

David Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see chapter 16.
Clark Kimberling, Geometry In Action, Key College Publishing, 2003, pages 115-116.

Michael Cavers, Spiked Math #524 (2012)
Clark Kimberling, Geometry In Action, 2003, scanned copy (with permission). See pages 115-116.
Index entries for algebraic numbers, degree 2.

Cf. A002163, A010466, A195285, A195304.

philo := proc(a,b,c) local f, A, B, C, P:
f:=2*a*b/(a+b+c):
A:=f*sqrt((a^2+(b+c)^2))/(b+c):
B:=f*sqrt((b^2+(c+a)^2))/(c+a):
C:=f*sqrt(2):
P:=(A+B+C)/(a+b+c):
print(simplify([A,B,C,P])):
print(evalf([A,B,C,P])): end:
philo(3,4,5); # Georg Fischer, Jul 18 2021

a = 3; b = 4; c = 5;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c); (* incenter *)
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) 195284 *)
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]
f2 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100] (* (B) A002163 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (C) A010466 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I) A195285 *)

(2/3)*sqrt(10) \\ Michel Marcus, Dec 24 2017

Equals (2/3)*sqrt(10).

Table and formulas corrected by Georg Fischer, Jul 17 2021

A020765 Decimal expansion of 1/sqrt(8).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Multiplied by 10, this is the real and the imaginary part of sqrt(25i). - Alonso del Arte, Jan 11 2013
Radius of the midsphere (tangent to the edges) in a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013
The side of the largest cubical present that can be wrapped (with cutting) by a unit square of wrapping paper. See Problem 10716 link. - Michel Marcus, Jul 24 2018
The ratio between the thickness and diameter of a geometrically fair coin having an equal probability, 1/3, of landing on each of its two faces and on its side after being tossed in the air. The calculation is based on comparing the areal projections of the faces and sides of the coin on a circumscribing sphere. (Mosteller, 1965). See A020760 for a physical solution. - Amiram Eldar, Sep 01 2020

			1/sqrt(8) = 0.353553390593273762200422181052424519642417968844237018294...

Frederick Mosteller, Fifty challenging problems of probability, Dover, New York, 1965. See problem 38, pp. 10 and 58-60.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Michael L. Catalano-Johnson, Daniel Loeb and John Beebee, A cubical gift: Problem 10716, The American Mathematical Monthly, Vol. 108, No. 1 (Jan., 2001), pp. 81-82.
Wikipedia, Tetrahedron.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2

Cf. Midsphere radii in Platonic solids:
A020761 (octahedron),
A010503 (cube),
A019863 (icosahedron),
A239798 (dodecahedron).

Digits:=100; evalf(1/sqrt(8)); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[N[1/Sqrt[8], 200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)
realDigitsRecip[Sqrt[8]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Apr 05 2025 *)

sqrt(1/8) \\ Charles R Greathouse IV, Apr 25 2016

A010503 divided by 2.
Equals A201488 minus 1/2. Equals 1/(A010487-4) minus 1/4. - Jon E. Schoenfield, Jan 09 2017
Equals Integral_{x=0..oo} x*exp(-x)*BesselJ(0,x) dx. - Kritsada Moomuang, Jun 03 2025

A378394 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a deltoidal icositetrahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 30 2024

The deltoidal icositetrahedron is the dual polyhedron of the (small) rhombicuboctahedron.

			2.410613141653407606153665785465949185980362906...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Deltoidal icositetrahedron.
Index entries for transcendental numbers.

Cf. A378390 (surface area), A378391 (volume), A378392 (inradius), A378393 (midradius).

Cf. A177870 and A195702 (dihedral angles of a (small) rhombicuboctahedron).

First[RealDigits[ArcSec[Sqrt[32] - 7], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["DeltoidalIcositetrahedron", "DihedralAngles"]], 10, 100]]

acos(-(4*sqrt(2) + 7)/17) \\ Charles R Greathouse IV, Feb 11 2025

Equals arcsec(4*sqrt(2) - 7) = arcsec(A010487 - 7).

Equals arccos(-(4*sqrt(2) + 7)/17) = arccos(-(A010487 + 7)/17).

A378351 Decimal expansion of the surface area of a (small) triakis octahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 23 2024

The (small) triakis octahedron is the dual polyhedron of the truncated cube.

			10.672941873983546705150008922490160564590104237715...

Eric Weisstein's World of Mathematics, Small Triakis Octahedron.
Wikipedia, Triakis octahedron.

Cf. A378352 (volume), A378353 (inradius), A201488 (midradius), A378354 (dihedral angle).
Cf. A377298 (surface area of a truncated cube with unit edge).
Cf. A010487.

First[RealDigits[3*Sqrt[7 + Sqrt[32]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisOctahedron", "SurfaceArea"], 10, 100]]

Equals 3*sqrt(7 + 4*sqrt(2)) = 3*sqrt(7 + A010487).

A377342 Decimal expansion of the volume of a truncated octahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			11.3137084989847603904135097936775846285573750030...

Eric Weisstein's World of Mathematics, Truncated Octahedron.
Wikipedia, Truncated octahedron.

Cf. A377341 (surface area), A020797 (circumradius/10), A152623 (midradius).
Cf. A131594 (analogous for a regular octahedron).
Cf. A002193, A010466, A010487.

First[RealDigits[8*Sqrt[2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedOctahedron", "Volume"], 10, 100]]

Equals 8*sqrt(2) = 8*A002193 = 4*A010466 = 2*A010487.

A378353 Decimal expansion of the inradius of a (small) triakis octahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 23 2024

The (small) triakis octahedron is the dual polyhedron of the truncated cube.

			0.81914066340325716171549134573565316624155520306132...

Cf. A378351 (surface area), A378352 (volume), A201488 (midradius), A378354 (dihedral angle).

Cf. A010487.

First[RealDigits[Sqrt[23/68 + Sqrt[32]/17], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisOctahedron", "Inradius"], 10, 100]]

Equals sqrt(23/68 + 4*sqrt(2)/17) = sqrt(23/68 + A010487/17).

A041053 Denominators of continued fraction convergents to sqrt(32).

Original entry on oeis.org

1, 1, 2, 3, 32, 35, 67, 102, 1087, 1189, 2276, 3465, 36926, 40391, 77317, 117708, 1254397, 1372105, 2626502, 3998607, 42612572, 46611179, 89223751, 135834930, 1447573051, 1583407981, 3030981032, 4614389013

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 (0,0,0,34,0,0,0,-1).

Cf. A010487, A041052.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[32],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)
Denominator[Convergents[Sqrt[32],30]] (* or *) LinearRecurrence[{0,0,0,34,0,0,0,-1},{1,1,2,3,32,35,67,102},30] (* Harvey P. Dale, Jun 15 2015 *)

G.f.: (1+x+2*x^2+3*x^3-2*x^4+x^5-x^6)/(1-34*x^4+x^8). [Colin Barker, Mar 13 2012]

a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=32, a(5)=35, a(6)=67, a(7)=102, a(n)=34*a(n-4)-a(n-8). - Harvey P. Dale, Jun 15 2015

A195286 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(5,12,13).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 14 2011

See A195284 for definitions and a general discussion.

			(A)=4.0792156108742278640225792872182255...

Cf. A195284, A195288, A195289.

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 *)

A195288 Decimal expansion of shortest length, (C), of segment from side CA through incenter to side CB in right triangle ABC with sidelengths (a,b,c)=(5,12,13).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 14 2011

See A195284 for definitions and a general discussion.

			(C)=4.80740170061865239082562835...

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 *)

Showing 1-10 of 24 results. Next

cp's OEIS Frontend

A248259 Egyptian fraction representation of sqrt(32) (A010487) using a greedy function.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A195284 Decimal expansion of shortest length of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5); i.e., decimal expansion of 2*sqrt(10)/3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A020765 Decimal expansion of 1/sqrt(8).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A378394 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a deltoidal icositetrahedron.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A378351 Decimal expansion of the surface area of a (small) triakis octahedron with unit shorter edge length.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A377342 Decimal expansion of the volume of a truncated octahedron with unit edge length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A378353 Decimal expansion of the inradius of a (small) triakis octahedron with unit shorter edge length.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula