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 1-4 of 4 results.

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

A195496 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(B)=1.017153446754804466256798187816606336...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = b - 1; b = GoldenRatio; 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) A195495 *)
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) A195496 *)
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) A195497 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)

A195497 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			(C)=0.862968792141037434136010433016173...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = b - 1; b = GoldenRatio; 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) A195495 *)
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) A195496 *)
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) A195497 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)

A195498 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the right triangle ABC having sidelengths (a,b,c)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 19 2011

Keywords

Comments

See A195304 for definitions and a general discussion.

Examples

			Philo(ABC,G)=0.575915236513482373618787369187419918767270...

Crossrefs

Cf. A195304.

Programs

  • Mathematica
    a = b - 1; b = GoldenRatio; 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) A195495 *)
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) A195496 *)
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) A195497 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)
Showing 1-4 of 4 results.

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