cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 94 results. Next

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).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 18 2011

Keywords

Comments

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.

Examples

			1.89630056630920201475386720365481991708010328....

Links

Crossrefs

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

Programs

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

A197008 Decimal expansion of the shortest distance from x axis through (1,2) to y axis.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P. Suppose that T is the angle formed by the positive x and y axes and that h>0 and k>0. Notation:
...
P=(h,k)
L=the Philo line of P across T
U=x-intercept of L
V=y-intercept of L
d=|UV|
...
Although Philo lines are not generally Euclidean-constructible, exact expressions for U, V, and d can be found for the angle T under consideration. Write u(t)=(t,0), let v(t) the corresponding point on the y axis, and let d(t) be the distance between u(t) and v(t). Then d is found by minimizing d(t)^2:
d=w*sqrt(1+(k/h)^(2/3)), where w=(h+(h*k^2))^(1/3).
...
Guide:
h....k...........d
1....2........A197008
1....3........A197012
1....4........A197013
2....3........A197014
3....4........A197015
1..sqrt(2)....A197031
...
For guides to other Philo lines, see A195284 and A197032.
The cube root of any positive number can be connected to the Philo lines (or Philon lines) for a 90-degree angle. If the equation x^3-2 is represented using Lill's method, it can be shown that the path of the root 2^(1/3) creates the shortest segment (Philo line) from the x axis through (1,2) to the y axis. For more details see the article "Lill's method and the Philo Line for Right Angles" linked below. - Raul Prisacariu, Apr 06 2024

Examples

			d=4.161938184941462752390080...
x-intercept: U=(2.5874..., 0)
y-intercept: V=(0, 3.2599...)

Links

Crossrefs

Cf. A197012, A005480, A195284.

Programs

  • Maple
    (1+2^(2/3))^(3/2); evalf(%) ; # R. J. Mathar, Nov 08 2022
  • Mathematica
    f[x_] := x^2 + (k*x/(x - h))^2; t = h + (h*k^2)^(1/3);
h = 1; k = 2; d = N[f[t]^(1/2), 100]
RealDigits[d] (* this sequence *)
x = N[t] (* x-intercept; -1+4^(1/3); cf. A005480 *)
y = N[k*t/(t - h)] (* y-intercept *)
Show[Plot[k + k (x - h)/(h - t), {x, 0, t}],
ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 4}], PlotRange -> All, AspectRatio -> Automatic]
  • PARI
    polrootsreal(x^6 - 15*x^4 - 33*x^2 - 125)[2] \\ Charles R Greathouse IV, Feb 03 2025

A197032 Decimal expansion of the x-intercept of the shortest segment from the positive x axis through (2,1) to the line y=x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T. For discussions and guides to related sequences, see A197008 and A195284.
Philo lines from positive x axis through (h,k) to line y=mx:
m......h......k....x-intercept.....distance
1......2......1.......A197032......A197033
1......3......1.......A197034......A197035
1......4......1.......A197136......A197137
1......3......2.......A197138......A197139
2......1......1.......A197140......A197141
2......2......1.......A197142......A197143
2......3......1.......A197144......A197145
2......4......1.......A197146......A197147
3......1......1.......A197148......A197149
3......2......1.......A197150......A197151
1/2....3......1.......A197152......A197153
1/2....4......1.......A197154......A197155

Examples

			length of Philo line:  1.8442716817001... (see A197033)
endpoint on x axis: (2.35321..., 0)
endpoint on y=x:    (1.73898, 1.73898)

Links

Crossrefs

Cf. A197033, A197008, A195284.
Cf. A357469 (= this constant - 1).

Programs

  • Maple
    Digits := 140 ;
x^3-4*x^2+6*x-5 ;
fsolve(%=0) ; # R. J. Mathar, Nov 08 2022
  • Mathematica
    f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
g[t_] := D[f[t], t]; Factor[g[t]]
p[t_] := h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 - 3 h m t^2 + m t^3 (* root of p[t] minimizes f *)
m = 1; h = 2; k = 1; (* m=slope; (h,k)=point *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197032 *)
{N[t], 0} (* lower endpoint of minimal segment [Philo line] *)
{N[k*t/(k + m*t - m*h)],
N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
d = N[Sqrt[f[t]], 100]
RealDigits[d] (* A197033 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2.5}],
ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 3}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]
  • PARI
    solve(x=2,3, x^3 - 4*x^2 + 6*x - 5)

Formula

x = 2 + tan phi where 1 + 2 tan phi = 1/(sin phi + cos phi), whence x = 1 + A357469 = the only real root of x^3 - 4*x^2 + 6*x - 5. - M. F. Hasler, Nov 08 2022

Extensions

Invalid trailing digits corrected by R. J. Mathar, Nov 08 2022

A010487 Decimal expansion of square root of 32.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Continued fraction expansion is 5 followed by {1, 1, 1, 10} repeated. - Harry J. Smith, Jun 04 2009
Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(5,12,13), see A195284. - Clark Kimberling, Sep 14 2011
Length of the perimeter of a square with circumscribed unit circle. - R. J. Mathar, Aug 24 2023

Examples

			5.656854249492380195206754896838792314278687501507792292706718951962929....

Links

Crossrefs

Cf. A010130 (continued fraction).

Programs

  • Mathematica
    RealDigits[N[Sqrt[32],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)
  • PARI
    default(realprecision, 20080); x=sqrt(32); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010487.txt", n, " ", d));  \\ Harry J. Smith, Jun 04 2009

Formula

Equals 4 * A002193. - R. J. Mathar, Oct 16 2015

A010524 Decimal expansion of square root of 72.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

This is also the ratio of the volume of a cube to the volume of a regular tetrahedron of the same edge length. - Rick L. Shepherd, May 29 2002
Continued fraction expansion is 8 followed by {2, 16} repeated. - Harry J. Smith, Jun 08 2009
Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(8,15,17), see A195284. - Clark Kimberling, Sep 14 2011
Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(7,24,25). - Clark Kimberling, Sep 14 2011

Examples

			8.485281374238570292810132345258188471418031252261688439060078427944394...

Links

Crossrefs

Cf. A040063 (continued fraction), A002193, A195284.

Programs

  • Mathematica
    RealDigits[N[72^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 23 2012 *)
  • PARI
    default(realprecision, 20080); x=sqrt(72); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010524.txt", n, " ", d));  \\ Harry J. Smith, Jun 08 2009

Formula

Equals 6*A002193. - Omar E. Pol, Mar 09 2021

A163960 Decimal expansion of 2*(sqrt(2) - 1).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Oct 02 2010

Keywords

Comments

Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)). (See A195284.) - Clark Kimberling, Sep 14 2011

Examples

			0.82842712474619009760337744841939615713934375075389614635335...

References

  • J. M. Steele, Probability Theory and Combinatorial Optimization, SIAM, 1997, p. 3.

Links

Crossrefs

Cf. A084158, A156035, A195284.
Essentially the same digit sequence as A010466, A086178, A090488 and A157258.

Programs

Formula

Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1) * 4^k). - Amiram Eldar, May 06 2022
Equals Sum_{k>=1} (-1)^(k+1)/A084158(k). - Amiram Eldar, Dec 02 2024

A195345 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,3,sqrt(10)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 16 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(B)=1.0326414912093473614756376556576114854142201858...

Links

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 1; b = 3; c = Sqrt[10]; f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]  (* A195344 *)
N[x2, 100]
RealDigits[%] (* A195345 *)
N[x3, 100]
RealDigits[%] (* A195346 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* A195347 *)
  • PARI
    (6*sqrt(20 + sqrt(40)))/(14 + 5*sqrt(10)) \\ Charles R Greathouse IV, Feb 11 2025

A195346 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)=(1,3,sqrt(10)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 16 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(C)=1.1847182944928008023884075593762398433974507...

Links

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 1; b = 3; c = Sqrt[10]; f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]  (* A195344 *)
N[x2, 100]
RealDigits[%] (* A195345 *)
N[x3, 100]
RealDigits[%] (* A195346 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* A195347 *)
  • PARI
    sqrt(72)/(sqrt(10)+4) \\ Charles R Greathouse IV, Feb 11 2025

A195347 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,3,sqrt(10) right triangle ABC.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 16 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			Philo(ABC,I)=0.4280818058125219350252671517036980901568443655...

Links

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 1; b = 3; c = Sqrt[10]; f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%] (* A195344 *)
N[x2, 100]
RealDigits[%] (* A195345 *)
N[x3, 100]
RealDigits[%] (* A195346 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* A195347 *)
  • PARI
    polrootsreal(6561*x^8 - 42107688*x^6 + 495305280*x^5 + 39826979224*x^4 - 60652800*x^3 - 4964068512*x^2 - 900806400*x - 44270064)[6] \\ Charles R Greathouse IV, Feb 11 2025

Extensions

a(99) corrected by Georg Fischer, Jul 18 2021

A195340 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)=(1,2,sqrt(5)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 16 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(A)=0.7849296847644349452017243634567...

Links

Crossrefs

Cf. A195284, A195341, A195342, A195343.

Programs

  • Mathematica
    a = 1; b = 2; c = Sqrt[5]; f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ];
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ];
x3 = f*Sqrt[2];
N[x1, 100]
RealDigits[%] (* (A) A195340 *)
N[x2, 100]
RealDigits[%] (* (B) A195341 *)
N[x3, 100]
RealDigits[%] (* (C) A195342 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* Philo(ABC,I) A195343 *)
  • PARI
    polrootsreal(x^4 - 520*x^2 + 320)[3] \\ Charles R Greathouse IV, Feb 11 2025
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