A010466 -id:A010466 - 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

A086178 Decimal expansion of 1 + 2*sqrt(2).

Original entry on oeis.org

3, 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, 4, 7

Offset: 1

Eric W. Weisstein, Jul 11 2003

Onset of 3-cycle in the logistic equation.

			3.8284...

J. Bechhoefer, The Birth of Period 3, Revisited, Mathematics Magazine pp. 115-8 Vol. 69 No. Apr 02 1996
W. B. Gordon, Period Three Trajectories of the Logistic Map, Mathematics Magazine pp. 118-120 Vol. 69 No. Apr 02 1996
J. Meiss, Period three saddle-node bifurcation for the Logistic Map
P. Saha & S. H. Strogatz, The Birth of period 3, Mathematics Magazine pp. 42-7 Vol. 68, No. 1, 1995.
Eric Weisstein's World of Mathematics, Logistic Map
Index entries for algebraic numbers, degree 2

Cf. A086179.

RealDigits[1+2Sqrt[2],10,120][[1]] (* Harvey P. Dale, Jun 13 2013 *)

1+2*sqrt(2) \\ Felix Fröhlich, Sep 26 2017

Equals 1+A010466 . - R. J. Mathar, Sep 12 2008

A090488 Decimal expansion of 2 + 2*sqrt(2).

Original entry on oeis.org

4, 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, 4

Offset: 1

Felix Tubiana, Feb 05 2004

Side length of smallest square containing five circles of radius 1. - Charles R Greathouse IV, Apr 05 2011
Equals n + n/(n +n/(n +n/(n +....))) for n = 4.  See also A090388. - Stanislav Sykora, Jan 23 2014
Also the area of a regular octagon with unit edge length. - Stanislav Sykora, Apr 12 2015
The positive solution to x^2 - 4*x - 4 = 0. The negative solution is -1 * A163960 = -0.82842... . - Michal Paulovic, Dec 12 2023

			4.828427124746190097603377448419396157139343750...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Octagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 2

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Cf. A002193, A010466, A014176, A086178, A156035, A157258, A163960, A365823.
Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A256853, A178816, A256854, A178809.

RealDigits[2+2Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 11 2015 *)

2*(1 + sqrt(2)) \\ G. C. Greubel, Jul 03 2017

Equals 1 + A086178 = 2*A014176. - R. J. Mathar, Sep 03 2007
From Michal Paulovic, Dec 12 2023: (Start)
Equals A010466 + 2.
Equals A156035 - 1.
Equals A157258 - 5.
Equals A163960 + 4.
Equals A365823 - 2.
Equals [4; 1, 4, ...] (periodic continued fraction expansion).
Equals sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * ...)))). (End)

Better definition from Rick L. Shepherd, Jul 02 2004

A022842 Beatty sequence for sqrt(8).

Original entry on oeis.org

2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 31, 33, 36, 39, 42, 45, 48, 50, 53, 56, 59, 62, 65, 67, 70, 73, 76, 79, 82, 84, 87, 90, 93, 96, 98, 101, 104, 107, 110, 113, 115, 118, 121, 124, 127, 130, 132, 135, 138, 141, 144, 147, 149, 152, 155, 158, 161, 164

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Index entries for sequences related to Beatty sequences

A bisection of A001951. Cf. A010466.

[Floor(n*Sqrt(8)): n in [1..60]]; // Vincenzo Librandi, Oct 24 2011

a:=n->floor(2*n*sqrt(2)): seq(a(n),n=1..60); # Muniru A Asiru, Sep 28 2018

Table[Floor[2*n*Sqrt[2]], {n,1,60}] (* G. C. Greubel, Sep 28 2018 *)

vector(80, n, floor(2*n*sqrt(2))) \\ G. C. Greubel, Sep 28 2018

from sympy import integer_nthroot
def A022842(n): return integer_nthroot(8*n**2,2)[0] # Chai Wah Wu, Mar 16 2021

a(n) = floor(2*n*sqrt(2)). - Michel Marcus, Oct 31 2017

Offset changed from 0 to 1 by Vincenzo Librandi, Oct 24 2011

A040005 Continued fraction for sqrt(8).

Original entry on oeis.org

2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4

Offset: 0

N. J. A. Sloane

			2.828427124746190097603377448... = 2 + 1/(1 + 1/(4 + 1/(1 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 02 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010466 (decimal expansion).

Essentially the same as A010685.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[8],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

{ allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(8)); for (n=0, 20000, write("b040005.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2^e) = 4, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 3/2^s). (End)
G.f.: (2 + x + 2*x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A187110 Decimal expansion of sqrt(3/8).

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Apart from leading digits, the same as A174925.

Radius of the circumscribed sphere (congruent with vertices) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			sqrt(3/8)=0.61237243569579452454932101867647284799148687016417..

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

Michael Penn, Maximize the area of one ellipse!, YouTube video, 2020.
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010464, A010466, A020781, A040001, A040005, A115754, A301755.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron). - Stanislav Sykora, Feb 10 2014

Mathematica
```
RealDigits[N[Sqrt[3/8],300]][[1]]
```

sqrt(3/8) \\ Charles R Greathouse IV, Mar 16 2022

Equals A010464/4. - Stefano Spezia, Jan 26 2025

Equals 3*A020781 = A115754/2 = sqrt(A301755). - Hugo Pfoertner, Jan 26 2025

A041011 Denominators of continued fraction convergents to sqrt(8).

Original entry on oeis.org

1, 1, 5, 6, 29, 35, 169, 204, 985, 1189, 5741, 6930, 33461, 40391, 195025, 235416, 1136689, 1372105, 6625109, 7997214, 38613965, 46611179, 225058681, 271669860, 1311738121, 1583407981, 7645370045, 9228778026, 44560482149

Offset: 0

N. J. A. Sloane

Sqrt(8) = 2 + continued fraction [0; 1, 4, 1, 4, 1, 4, ...] = 4/2 + 4/5 + 4/(5*29) + 4/(29*169) + 4/(169*985) + ... - Gary W. Adamson, Dec 21 2007
This is the sequence of Lehmer numbers U_n(sqrt(R),Q) with the parameters R = 4 and Q = -1. It is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. - Peter Bala, May 12 2014
Apparently the same as A152118(n). - Georg Fischer, Jul 01 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [shifted by _Georg Fischer_, Jul 01 2019]
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Hongshen Chua, A Study of Second-Order Linear Recurrence Sequences via Continuants, J. Int. Seq. (2023) Vol. 26, Art. 23.8.8.
J. L. Ramirez and F. Sirvent, A q-Analogue of the Bi-Periodic Fibonacci Sequence, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=1, b=4.
Eric Weisstein's World of Mathematics, Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A000129, A010466, A041010. A089499, A136211, A152118.

I:=[1, 1, 5, 6]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

with(combinat): a := n -> fibonacci(n + 1, 2)/2^(n mod 2):
seq(a(n), n = 0 .. 28); # Miles Wilson, Aug 04 2024

Denominator[NestList[(4/(4 + #))&, 0, 60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
CoefficientList[Series[(x + x^2 - x^3)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2013 *)
a0[n_] := ((3-2*Sqrt[2])^n*(2+Sqrt[2])-(-2+Sqrt[2])*(3+2*Sqrt[2])^n)/4 // Simplify
a1[n_] := (-(3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/(4*Sqrt[2]) // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0,6,0,-1},{1,1,5,6},40] (* Harvey P. Dale, Oct 21 2024 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,6,0]^n*[1;1;5;6])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

my(x='x+O('x^99)); concat(0, Vec((1+x-x^2)/(1-6*x^2+x^4))) \\ Altug Alkan, Mar 27 2016

a(2n) = A000129(2n+1), a(2n+1) = A000129(2n+2)/2.
a(n) = 6*a(n-2) - a(n-4). Also:
a(2n) = a(2n-1)+a(2n-2), a(2n+1)=4*a(2n)+a(2n-1).
G.f.: (1+x-x^2)/(1-6*x^2+x^4).
From Peter Bala, May 12 2014: (Start)
For n even, a(n) = (alpha^n - beta^n)/(alpha - beta), and for n odd, a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2), where alpha = 1 + sqrt(2) and beta = 1 - sqrt(2).
a(n) = Product_{k = 1..floor((n-1)/2)} ( 4 + 4*cos^2(k*Pi/n) ) for n >= 1. (End)
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((3-2*sqrt(2))^n*(2+sqrt(2))-(-2+sqrt(2))*(3+2*sqrt(2))^n)/4.
a1(n) = (-(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/(4*sqrt(2)). (End)
a(n) = ((-(-1 - sqrt(2))^n - 3*(1-sqrt(2))^n + (-1+sqrt(2))^n + 3*(1+sqrt(2))^n))/(8*sqrt(2)). - Colin Barker, Mar 27 2016

Entry improved by Michael Somos

First term 0 in b-file, formulas and programs removed by Georg Fischer, Jul 01 2019

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

N. J. A. Sloane, Oct 02 2010

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

			0.82842712474619009760337744841939615713934375075389614635335...

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

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for algebraic numbers, degree 2.

Cf. A084158, A156035, A195284.

Essentially the same digit sequence as A010466, A086178, A090488 and A157258.

RealDigits[2(Sqrt[2]-1),10,120][[1]] (* Harvey P. Dale, May 27 2016 *)

2*(sqrt(2)-1) \\ G. C. Greubel, Aug 13 2017

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

Showing 1-10 of 69 results. Next

cp's OEIS Frontend

A248238 Egyptian fraction representation of sqrt(8) (A010466) using a greedy function.

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A156035 Decimal expansion of 3 + 2*sqrt(2).

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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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A086178 Decimal expansion of 1 + 2*sqrt(2).

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A090488 Decimal expansion of 2 + 2*sqrt(2).

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A022842 Beatty sequence for sqrt(8).

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A040005 Continued fraction for sqrt(8).

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