A120683 -id:A120683 - 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

A010469 Decimal expansion of square root of 12.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

3+sqrt(12) is the ratio of the radii of the three identical kissing circles to that of their inner Soddy circle. - Lekraj Beedassy, Mar 04 2006
sqrt(12)-3 = 2*sqrt(3)-3 is the area of the largest equilateral triangle that can be inscribed in a unit square (as stated in MathWorld/Weisstein link). - Rick L. Shepherd, Jun 24 2006
Continued fraction expansion is 3 followed by {2, 6} repeated (A040008). - Harry J. Smith, Jun 02 2009
Surface of a regular octahedron with unit edge, and twice the surface of a regular tetrahedron with unit edge. - Stanislav Sykora, Nov 21 2013
Imaginary part of the square of a complex cubic root of 64 (real part is -2). - Alonso del Arte, Jan 13 2014

			3.4641016151377545870548926830...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.31.4 and 2.31.5, pp. 201-202.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Equilateral Triangle.
Wikipedia, Octahedron.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A120683.
Cf. A040008 (continued fraction), A041016 (numerators of convergents), A041017 (denominators).
Cf. A002194 (surface of tetrahedron), A010527 (surface of icosahedron/10), A131595 (surface of dodecahedron).

evalf[100](sqrt(12)); # Muniru A Asiru, Feb 12 2019

RealDigits[N[Sqrt[12], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=sqrt(12); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010469.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

Equals 2*sqrt(3) = 2*A002194. - Rick L. Shepherd, Jun 24 2006

A019884 Decimal expansion of sine of 75 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the real part of i^(1/6). - Stanislav Sykora, Apr 25 2012

Length of one side of the new Type 15 Convex Pentagon. - Michel Marcus, Aug 04 2015

			0.96592582628906828674974319972889736763390483900840455040234307631042...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Casey Mann et al, Type 15 Convex Pentagon, Jul 29 2015.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 4

Cf. A120683.

RealDigits[Sin[75 Degree], 10, 100][[1]] (* Vincenzo Librandi, Aug 11 2014 *)

cos(Pi/12) \\ Charles R Greathouse IV, Apr 25 2012

Equals cos(Pi/12) = [1+sqrt(3)]/[2*sqrt(2)] = A090388 * A020765. - R. J. Mathar, Jun 18 2006
Equals A019859 * A019874 + A019834 * A019849 = A019881 * A019896 + A019812 * A019827 . - R. J. Mathar, Jan 27 2021
Equals 1/(sqrt(6) - sqrt(2)) = 1/A120683. - Amiram Eldar, Aug 04 2022
Largest of the 4 real-valued roots of 16*x^4 -16*x^2 +1=0. - R. J. Mathar, Aug 29 2025
4*this^3 -3*this = A010503. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/8,1/8 ;1/2;3/4) = 2F1(-1/6,1/6;1/2;1/2). - R. J. Mathar, Aug 31 2025

A101263 Decimal expansion of sqrt(2 - sqrt(3)), edge length of a regular dodecagon with circumradius 1.

Original entry on oeis.org

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

Offset: 0

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 25 2005

sqrt(2 - sqrt(3)) is the shape of the lesser sqrt(6)-contraction rectangle, as defined at A188739. - Clark Kimberling, Apr 16 2011
This is a constructible number, since 12-gon is a constructible polygon. See A003401 for more details. - Stanislav Sykora, May 02 2016
It is also smaller positive coordinate of (symmetrical) intersection points of x^2 + y^2 = 4 circle and y = 1/x hyperbola. The bigger coordinate is A188887. - Leszek Lezniak, Sep 18 2018
The greatest possible minimum distance between 8 points in a unit square (Schaer and Meir, 1965; Schaer, 1965; Croft et al., 1991). - Amiram Eldar, Feb 24 2025

			0.517638090205041524697797675248096656698137802639861027628006414630113....

Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

G. C. Greubel, Table of n, a(n) for n = 0..5000
J. Schaer and A. Meir, On a geometric extremum problem, Canadian Mathematical Bulletin, Vol. 8, No. 1 (1965), pp. 21-27.
J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
Eric Weisstein's World of Mathematics, Dodecagon.
Index entries for algebraic numbers, degree 4.

Cf. A003401, A019824, A019913, A091999, A120683, A188739, A188887, A329247.

r = 6^(1/2); t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]  (*A101263*)
RealDigits[Sqrt[2-Sqrt[3]],10,120][[1]] (* Harvey P. Dale, Apr 24 2018 *)

2*sin(Pi/12) \\ Stanislav Sykora, May 02 2016

Equals sqrt(A019913). - R. J. Mathar, Apr 20 2009
Equals 2*sin(Pi/12) = 2*cos(Pi*5/12). - Stanislav Sykora, May 02 2016
Equals i^(5/6) + i^(-5/6). - Gary W. Adamson, Jul 07 2022
From Amiram Eldar, Nov 24 2024: (Start)
Equals A120683 / 2 = 2 * A019824 = 1 / A188887 = exp(-A329247).
Equals (sqrt(3)-1)/sqrt(2).
Equals Product_{k>=1} (1 + (-1)^k/A091999(k)). (End)

A092242 Numbers that are congruent to {5, 7} (mod 12).

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 41, 43, 53, 55, 65, 67, 77, 79, 89, 91, 101, 103, 113, 115, 125, 127, 137, 139, 149, 151, 161, 163, 173, 175, 185, 187, 197, 199, 209, 211, 221, 223, 233, 235, 245, 247, 257, 259, 269, 271, 281, 283, 293, 295, 305, 307, 317, 319, 329, 331

Offset: 1

Giovanni Teofilatto, Feb 19 2004

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 64.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Fifth row of A092260.

Cf. A010527, A120683.

Select[Range[331], MemberQ[{5, 7}, Mod[#, 12]] &] (* Amiram Eldar, Dec 04 2021 *)

1/5^2 + 1/7^2 + 1/17^2 + 1/19^2 + 1/29^2 + 1/31^2 + ... = Pi^2*(2 - sqrt(3))/36 = 0.073459792... [Jolley] - Gary W. Adamson, Dec 20 2006
a(n) = 12*n - a(n-1) - 12 (with a(1)=5). - Vincenzo Librandi, Nov 16 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 6*n - 3 - 2*(-1)^n.
G.f.: x*(5+2*x+5*x^2) / ( (1+x)*(x-1)^2 ). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2 - sqrt(3))*Pi/12. - Amiram Eldar, Dec 04 2021
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/12) (A120683).
Product_{n>=1} (1 + (-1)^n/a(n)) = (sqrt(3)/2)*sec(Pi/12) (= A010527 * A120683). (End)

Edited and extended by Ray Chandler, Feb 21 2004

A195348 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,sqrt(3),2) and vertex angles of degree measure 30,60,90.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(A)=0.7578747639260239988121867474270095303467925401944...
(A)=(4*sqrt(6-3*sqrt(3)))/(3+sqrt(3))
(B)=2-(2/3)sqrt(3)
(C)=sqrt(6)-sqrt(2)

Cf. A195284, A093821, A120683, A195380.

a = 1; b = Sqrt[3]; c = 2;
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) A195348 *)
N[x2, 100]
RealDigits[%] (* (B) A093821 *)
N[x3, 100]
RealDigits[%] (* (C) A120683 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* A195380 *)

A195380 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,sqrt(3),sqrt(1) right triangle ABC (angles 30, 60, 90).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			Philo(ABC,I)=0.55757017691709380372112914604292318...

Cf. A195284.

a = 1; b = Sqrt[3]; c = 2;
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) A195348 *)
N[x2, 100]
RealDigits[%] (* (B) A093821 *)
N[x3, 100]
RealDigits[%] (* (C) A120683 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%] (* A195380 *)

A281065 Decimal expansion of the greatest minimum separation between ten points in a unit square.

Original entry on oeis.org

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

Offset: 0

Jeremy Tan, Jan 14 2017

The corresponding values for two to nine points have simple expressions:
  N ... d_min
  2 ... sqrt(2)            (A002193)
  3 ... sqrt(6) - sqrt(2)  (A120683)
  4 ... 1                  (A000007)
  5 ... sqrt(2) / 2        (A010503)
  6 ... sqrt(13) / 6       (A381485)
  7 ... 4 - 2*sqrt(3)      (A379338)
  8 ... sqrt(2 - sqrt(3))  (A101263)
  9 ... 1 / 2              (A020761)
In contrast, the value for ten points has a minimal polynomial of degree 18.
The smallest square ten unit circles will fit into has side length s = 2 + 2/d = 6.74744152... and the maximum radius of ten non-overlapping circles in the unit square is 1 / s = 0.14820432...

			0.421279543983903432768821760650298...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

C. de Groot, R. Peikert, and D. Würtz, The Optimal Packing of Ten Equal Circles in a Square, IPS Research Report, ETH Zürich, No. 90-12, August 1990.
Eckard Specht, The best known packings of equal circles in a square.
Jeremy Tan, Sympy (Python) program.
Index entries for algebraic numbers, degree 18.

Cf. A281115 (10 points in unit circle), A000007, A002193, A010503, A020761, A101263, A120683, A379338, A381485.

RealDigits[x /. FindRoot[x^Range[18, 0, -1].{1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200}, {x, 2/5}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Feb 24 2025 *)

my(p = Pol([1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200])); polrootsreal(p)[1]

d is the smallest real root of 1180129*d^18 - 11436428*d^17 + 98015844*d^16 - 462103584*d^15 + 1145811528*d^14 - 1398966480*d^13 + 227573920*d^12 + 1526909568*d^11 - 1038261808*d^10 - 2960321792*d^9 + 7803109440*d^8 - 9722063488*d^7 + 7918461504*d^6 - 4564076288*d^5 + 1899131648*d^4 - 563649536*d^3 + 114038784*d^2 - 14172160*d + 819200.

A381485 Decimal expansion of sqrt(13)/6.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 24 2025

The greatest possible minimum distance between 6 points in a unit square.

The solution was found by Ronald L. Graham and reported by Schaer (1965).

			0.60092521257733154885320354457841599104188276230754...

Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
D. Würtz, M. Monagan, and R. Peikert, The history of packing circles in a square, Maple Technical Newsletter, Vol. 1 (1994), pp. 35-42; ResearchGate link.
Index entries for algebraic numbers, degree 2.

Solutions for k points: A002193 (k = 2), A120683 (k = 3), 1 (k = 4), A010503 (k = 5), this constant (k = 6), A379338 (k = 7), A101263 (k = 8), A020761 (k = 9), A281065 (k = 10).

Cf. A010470, A142464, A176019, A295330, A344069.

Mathematica
```
RealDigits[Sqrt[13] / 6, 10, 120][[1]]
```

list(len) = digits(floor(10^len*quadgen(52)/6));

Equals A010470 / 6 = A295330 / 3 = A344069 / 2 = A176019 - 1/2 = sqrt(A142464).

Minimal polynomial: 36*x^2 - 13.

Showing 1-9 of 9 results.

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.

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A010469 Decimal expansion of square root of 12.

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A019884 Decimal expansion of sine of 75 degrees.

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A101263 Decimal expansion of sqrt(2 - sqrt(3)), edge length of a regular dodecagon with circumradius 1.

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A092242 Numbers that are congruent to {5, 7} (mod 12).

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A195348 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,sqrt(3),2) and vertex angles of degree measure 30,60,90.

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A195380 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,sqrt(3),sqrt(1) right triangle ABC (angles 30, 60, 90).

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