A002163 -id:A002163 - cp's OEIS frontend

A386241 Decimal expansion of sqrt(5)*sin(Pi/8).

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

Offset: 0

Hugo Pfoertner, Jul 18 2025

Upper bound of the wobbling distance S of two rotated square lattices. See A307110 and A307731 for the special case of rotation angle Pi/4. According to Jan Fricke (1999), the angle Pi/4 is the most unfavorable case, i.e., smaller bounds can be found for all other angles.

			0.8557061686312838477748180718246837073...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Hans-Georg Carstens, Walter A. Deuber, Wolfgang Thumser, and Elke Koppenrade, Geometrical Bijections in Discrete Lattices. Combinatorics, Probability and Computing, 8(1-2), 109-129, 1999.
Jan Fricke, Symmetric Graphs have symmetric Matchings, arXiv:1607.07426 [math.GR], 25 Jul 2016, Introduction.
Klaus Nagel, A Lower Bound for Double Lattice Bijections, August 17, 2025.
Index entries for algebraic numbers, degree 4

Cf. A002163, A144981, A182168, A307110, A307731, A367150.

First[RealDigits[Sqrt[5]*Sin[Pi/8],10,100]] (* Paolo Xausa, Aug 13 2025 *)

sqrt(5)*sin(Pi/8) \\ Charles R Greathouse IV, Aug 19 2025

Equals A002163*A182168.

The minimal polynomial is 8*x^4 - 40*x^2 + 25. - Joerg Arndt, Aug 02 2025

A386852 Decimal expansion of the dihedral angle, in radians, between the pentagonal face and a triangular face in a pentagonal pyramid with equal edges (Johnson solid J_2).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Aug 05 2025

Also the dihedral angle, in radians, between the 10-gonal face and a triangular face in a pentagonal cupola (Johnson solid J_5)

			0.65235813978436818599539063164382257436530791996...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Pentagonal cupola.
Wikipedia, Pentagonal pyramid.
Index entries for transcendental numbers.

Cf. A179552 (J_2 volume), A179553 (J_2 surface area).
Cf. A179590 (J_5 volume), A179591 (J_5 surface area).
Cf. A002163, A010476.

First[RealDigits[ArcSec[Sqrt[15 - 6*Sqrt[5]]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J2", "DihedralAngles"]], 10, 100]]

acos(sqrt((5+2*sqrt(5))/15)) \\ Charles R Greathouse IV, Aug 19 2025

Equals arcsec(sqrt(15 - 6*sqrt(5))) = arcsec(sqrt(15 - 6*A002163)).

Equals arccos(sqrt((5 + 2*sqrt(5))/15)) = arccos(sqrt((5 + A010476)/15)).

A004558 Expansion of sqrt(5) in base 5.

Original entry on oeis.org

2, 1, 0, 4, 2, 2, 3, 2, 4, 0, 1, 1, 3, 2, 4, 1, 0, 4, 0, 0, 1, 3, 4, 4, 1, 2, 3, 3, 0, 4, 1, 3, 0, 4, 2, 4, 2, 2, 2, 1, 2, 1, 3, 2, 1, 1, 3, 0, 1, 3, 1, 0, 3, 2, 1, 0, 0, 1, 0, 2, 2, 1, 4, 2, 3, 4, 4, 4, 3, 4, 3, 4, 2, 4, 2, 3, 4, 4, 1, 4, 4, 4, 2, 1, 1, 3, 0, 3, 2, 1, 4, 0, 1, 3, 0, 3, 2, 2, 1

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1.2000 from Vincenzo Librandi)
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A002163.

Prune(Reverse(IntegerToSequence(Isqrt(5*5^200), 5))); // Vincenzo Librandi, Apr 29 2017

RealDigits[Sqrt[5], 5, 100][[1]] (* Vincenzo Librandi, Apr 29 2017 *)

Updated by Alois P. Heinz at the suggestion of Kevin Ryde, Feb 19 2012

A059176 Engel expansion of sqrt(5) = 2.23606...

Original entry on oeis.org

1, 1, 5, 6, 13, 16, 16, 38, 48, 58, 104, 177, 263, 332, 389, 4102, 4575, 5081, 9962, 18316, 86613, 233239, 342534, 964372, 1452850, 7037119, 7339713, 8270361, 12855437, 15900982, 19211148, 1365302354, 1565752087, 1731612283

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

Simon Plouffe, Table of n, a(n) for n = 1..883
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A002163.

Essentially the same as A028259.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[Sqrt[5], 7!], 50] (* modified by G. C. Greubel, Dec 26 2016 *)

A171546 Decimal expansion of sqrt(3/35).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 3/2 ; 0 -1/2 | 5/2 -1/2>.

			sqrt(3/35) = sqrt(105)/35 = 0.2927700218845599538063153..

Wikipedia, Table of Clebsch-Gordan coefficients

RealDigits[Sqrt[3/35],10,120][[1]]  (* Harvey P. Dale, Apr 20 2011 *)

equals A002194/A010490 = A171537/A002163 = A171539/A002193.

A176445 Decimal expansion of sqrt(1295).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of sqrt(1295) is 35 followed by (repeat 1, 70).

sqrt(1295) = sqrt(5)*sqrt(7)*sqrt(37).

			sqrt(1295) = 35.98610843089316319412...

Cf. A002163, A010465, A010491, A176444.

RealDigits[Sqrt[1295],10,120][[1]] (* Harvey P. Dale, Apr 19 2019 *)

A176909 Decimal expansion of sqrt(230).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 28 2010

Continued fraction expansion of sqrt(230) is 15 followed by A165734.

			sqrt(230) = 15.16575088810310110851...

Cf. A002193 (decimal expansion of sqrt(2)), A002163 (decimal expansion of sqrt(5)), A010479 (decimal expansion of sqrt(23)), A176906 (decimal expansion of (15+sqrt(230))/5), A165734 (repeat 6, 30).

A178039 Decimal expansion of sqrt(44310).

Original entry on oeis.org

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

Offset: 3

Klaus Brockhaus, May 17 2010

Continued fraction expansion of sqrt(44310) is 210 followed by (repeat 2, 420).

sqrt(44310) = sqrt(2)*sqrt(3)*sqrt(5)*sqrt(7)*sqrt(211).

			sqrt(44310) = 210.49940617493437480150...

Cf. A002193 (decimal expansion of sqrt(2)), A002194 (decimal expansion of sqrt(3)), A002163 (decimal expansion of sqrt(5)), A010465 (decimal expansion of sqrt(7)), A178040 (decimal expansion of sqrt(211)), A178038 (decimal expansion of (161+sqrt(44310))/259).

A195285 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 14 2011

See A195284 for a definition of Philo(ABC,I) and general discussion.

			Philo(ABC,I)=0.59772335075207494572...

Cf. A002163, A010466, A195284.

a = 3; b = 4; c = 5;
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) A195284 *)
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) A002163 *)
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) A010466 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I) A195285 *)

A229780 Decimal expansion of (3+sqrt(5))/10.

Original entry on oeis.org

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

Offset: 0

Joost Gielen, Sep 29 2013

sqrt((3+sqrt(5))/10) = sqrt(phi^2/5) = (5+sqrt(5))/10 = (3+sqrt(5))/10 + 2/10 = 0.723606797... .
Essentially the same as A134972, A134945, A098317 and A002163. - R. J. Mathar, Sep 30 2013
Equals one tenth of the limit of (G(n+2)+G(n+1)+G(n-1)+G(n-2))/G(n), where G(n) is any nonzero sequence satisfying the recurrence G(n+1) = G(n) + G(n-1) including A000032 and A000045, as n --> infinity. - Richard R. Forberg, Nov 17 2014
3+sqrt(5) is the perimeter of a golden rectangle with a unit width. - Amiram Eldar, May 18 2021
Constant x such that x = sqrt(x) - 1/5. - Andrea Pinos, Jan 15 2024

			0.5236067977499...

Cf. A094874, A187798.

RealDigits[GoldenRatio^2/5,10,120][[1]] (* Harvey P. Dale, Dec 02 2014 *)

(3+sqrt(5))/10 = (phi/sqrt(5))^2 = phi^2/5 where phi is the golden ratio.

cp's OEIS Frontend

A386241 Decimal expansion of sqrt(5)*sin(Pi/8).

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A386852 Decimal expansion of the dihedral angle, in radians, between the pentagonal face and a triangular face in a pentagonal pyramid with equal edges (Johnson solid J_2).

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A004558 Expansion of sqrt(5) in base 5.

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A059176 Engel expansion of sqrt(5) = 2.23606...

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A171546 Decimal expansion of sqrt(3/35).

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A176445 Decimal expansion of sqrt(1295).

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A176909 Decimal expansion of sqrt(230).

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A178039 Decimal expansion of sqrt(44310).

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A195285 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

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