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 18 results. Next

A248296 Egyptian fraction representation of sqrt(72) (A010524) using a greedy function.

Original entry on oeis.org

8, 3, 7, 111, 12212, 421899134, 214366287730447196, 74154301233407587376512952938963737, 22082353211860579770417392785370193807657413641357962334630621172698141
Offset: 0

Views

Author

Robert G. Wilson v, Oct 04 2014

Keywords

Crossrefs

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.
Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Programs

  • Mathematica
    Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 72]]

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.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

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.

Examples

			2.10818510677891955466592902962...

References

  • 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.

Links

Crossrefs

Cf. A002163, A010466, A195285, A195304.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    (2/3)*sqrt(10) \\ Michel Marcus, Dec 24 2017

Formula

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

Extensions

Table and formulas corrected by Georg Fischer, Jul 17 2021

A020829 Decimal expansion of 1/sqrt(72) = 1/(3*2^(3/2)) = sqrt(2)/12.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Volume of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
In the dragon curve fractal, (5/6)*sqrt(2) = 1.1785.... is the maximum distance of any point from curve start. Such a maximum must be to a vertex of the convex hull. Hull vertices are shown by Benedek and Panzone (theorem 3, page 85) and their P8 = 7/6 - (1/6)i at distance sqrt((7/6)^2 + (1/6)^2) is the maximum. - Kevin Ryde, Nov 22 2019
With offset 1, volume of a triangular cupola (Johnson solid J_3) with unit edges. - Paolo Xausa, Aug 04 2025

Examples

			0.117851130197757920733474...

References

  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Links

Crossrefs

Cf. A131594 (regular octahedron volume), A102208 (regular icosahedron volume), A102769 (regular dodecahedron volume).
Cf. A010524, A020765, A020775, A378207.

Programs

Formula

Equals Integral_{x=0..Pi/4} sin(x)^2 * cos(x) dx. - Amiram Eldar, May 31 2021
Equals 1/A010524 = A020765/3 = A020775/2 = A378207/5. - Hugo Pfoertner, Jan 26 2025

A378207 Decimal expansion of the midradius of a triakis tetrahedron with unit shorter edge length.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Nov 21 2024

Keywords

Comments

The triakis tetrahedron is the dual polyhedron of the truncated tetrahedron.

Examples

			0.589255650988789603667370301754040866070696614740...

Links

Crossrefs

Cf. A378204 (surface area), A378205 (volume), A378206 (inradius), A378208 (dihedral angle).
Cf. A093577 (midradius of a truncated tetrahedron with unit edge).
Cf. A010524.

Programs

  • Mathematica
    First[RealDigits[5/Sqrt[72], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisTetrahedron", "Midradius"], 10, 100]]
  • PARI
    5/sqrt(72) \\ Charles R Greathouse IV, Feb 11 2025

Formula

Equals 5/(6*sqrt(2)) = 5/A010524.

A040063 Continued fraction for sqrt(72).

Original entry on oeis.org

8, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			8.4852813742385702928101323... = 8 + 1/(2 + 1/(16 + 1/(2 + 1/(16 + ...)))). - _Harry J. Smith_, Jun 08 2009

Links

Crossrefs

Cf. A010524 Decimal expansion. - Harry J. Smith, Jun 08 2009

Programs

  • Maple
    Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[72],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},120,{16,2}] (* Harvey P. Dale, May 06 2018 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 31000); x=contfrac(sqrt(72)); for (n=0, 20000, write("b040063.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 08 2009

A195293 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)=(8,15,17).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(A)=6.18465843842649082473211478396111...

Crossrefs

Cf. A195284, A010524, A195296, A195297.

Programs

  • Mathematica
    a = 8; b = 15; c = 17;
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) A195293 *)
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) A195296 *)
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) A010524 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I), A195297 *)

A195296 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)=(8,15,17).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(C)=6.99714227381436056504898345305469969182567800...

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 8; b = 15; c = 17;
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) A195293 *)
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) A195296 *)
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) A010524 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I), A195297 *)

A195297 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of an 8,15,17 right triangle ABC.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			Philo(ABC,I)=0.54167705216198554206478076455685009252411270...

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 8; b = 15; c = 17;
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) A195293 *)
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) A195296 *)
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) A010524 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I), A195297 *)

A195290 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)=(7,24,25).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			(A)=6.0609152673132644948643802466...

Crossrefs

Cf. A195284, A195292.

Programs

  • Mathematica
    a = 7; b = 24; c = 25;
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) A195290 *)
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)=7.5 *)
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) A010524 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I) A195292 *)

A195292 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 7,24,25 right triangle ABC.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion.

Examples

			Philo(ABC,I)=0.39368208288485419263704486771198536...

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 7; b = 24; c = 25;
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) A195290 *)
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)=7.5 *)
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) A010524 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I) A195292 *)
Showing 1-10 of 18 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.