A010494 -id:A010494 - cp's OEIS frontend

A248266 Egyptian fraction representation of sqrt(40) (A010494) using a greedy function.

6, 4, 14, 320, 571786, 469930223859, 260342286471149560589985, 110737149164265654381526929767261159120340941327, 13640751783742037895965317463353502238298025074840803034014381823166601709380037834476485770683

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

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[ 40]]

A020797 Decimal expansion of 1/sqrt(40).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

With offset 1, decimal expansion of sqrt(5/2). - Eric Desbiaux, May 01 2008
sqrt(5/2) appears as a coordinate in a degree-5 integration formula on 13 points in the unit sphere [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011
With offset 2, decimal expansion of sqrt(250). - Michel Marcus, Nov 04 2013
From Wolfdieter Lang, Nov 21 2017: (Start)
The regular continued fraction of 1/sqrt(40) = 1/(2*sqrt(10)) is [0; 6, 3, repeat(12, 3)], and the convergents are given by A(n-1)/B(n-1), n >= 0, with A(-1) = 0, A(n-1) = A041067(n) and B(-1) = 1, B(n-1) = A041066(n).
The regular continued fraction of sqrt(5/2) = sqrt(10)/2 is [1; repeat(1, 1, 2)], and the convergents are given in A295333/A295334.
sqrt(10)/2 is one of the catheti of the rectangular triangle with hypotenuse sqrt(13)/2 = A295330 and the other cathetus sqrt(3)/2 = A010527. This can be constructed from a regular hexagon inscribed in a circle with a radius of 1 unit. If the vertex V_0 has coordinates (x, y) = (1, 0) and the midpoint M_4 = (0, -sqrt(3)/2) then the point L = (sqrt(10)/2, 0) is obtained as intersection of the x-axis and a circle around M_4 with radius taken from the distance between M_4 and V_1 = (1/2, sqrt(3)/2) which is sqrt(13)/2. (End)

			1/sqrt(40) = 0.15811388300841896659994467722163592668597775696626084134287...
sqrt(5/2) = 1.5811388300841896659994467722163592668597775696626084134287...
sqrt(250) = 15.811388300841896659994467722163592668597775696626084134287...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
A. H. Stroud, D. Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Index entries for algebraic numbers, degree 2.

Cf. A010467 (sqrt(10)), A010527, A010494 (sqrt(40)), A041067/A041066, A295330, A295333/A295334.

RealDigits[N[1/Sqrt[40],200]] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2010 *)

1/sqrt(40) \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(40*x^2-1)[2] \\ Charles R Greathouse IV, Feb 04 2025

Equals Re(sqrt(5*i)/10) = Im(sqrt(5*i)/10). - Karl V. Keller, Jr., Sep 01 2020

Equals A010467/20. - R. J. Mathar, Feb 23 2021

A132677 Period 3: repeat [1, 2, -3].

Original entry on oeis.org

1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3

Offset: 0

Paul Curtz, Nov 15 2007

a(n) is proportional to its 6n-th differences.

Nonsimple continued fraction expansion of 1+sqrt(2/5) = 1.63245553... (see A010494). - R. J. Mathar, Mar 08 2012

Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Cf. A000748, A010494, A049347, A101544.

&cat [[1, 2, -3]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([1, 2, -3]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016
A132677 := proc(n)
    op(1+modp(n,3),[1,2,-3]) ;
end proc: # R. J. Mathar, Mar 14 2025

PadRight[{}, 100, {1, 2, -3}] (* Wesley Ivan Hurt, Jul 01 2016 *)

a(n)=[1,2,-3][1+n%3] \\ Jaume Oliver Lafont, Mar 24 2009

G.f.: (1+3*x)/(1+x+x^2). - Jaume Oliver Lafont, Mar 24 2009
a(n) = cos(2*Pi*n/3) + 5*sin(2*Pi*n/3)/sqrt(3). - R. J. Mathar, Oct 08 2011
a(n) + a(n-1) + a(n-2) = 0 for n > 1, a(n) = a(n-3) for n > 2. - Wesley Ivan Hurt, Jul 01 2016

A041066 Numerators of continued fraction convergents to sqrt(40).

Original entry on oeis.org

6, 19, 234, 721, 8886, 27379, 337434, 1039681, 12813606, 39480499, 486579594, 1499219281, 18477210966, 56930852179, 701647437114, 2161873163521, 26644125399366, 82094249361619, 1011775117738794

Offset: 0

N. J. A. Sloane

With a(-1) = 1, a(n-1) gives, for n >= 0, the denominator of the convergents to 1/sqrt(40) = 1/(2*sqrt(10)). - Wolfdieter Lang, Nov 21 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

Cf. A041067 (denominators), A010494, A020797 (1/sqrt(40)).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[40],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)
Numerator[Convergents[Sqrt[40], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
LinearRecurrence[{0,38,0,-1},{6,19,234,721},20] (* Harvey P. Dale, Jul 20 2024 *)

G.f.: -(x+2)*(x^2-8*x-3) / ((x^2-6*x-1)*(x^2+6*x-1)). - Colin Barker, Nov 04 2013

A041067 Denominators of continued fraction convergents to sqrt(40).

Original entry on oeis.org

1, 3, 37, 114, 1405, 4329, 53353, 164388, 2026009, 6242415, 76934989, 237047382, 2921503573, 9001558101, 110940200785, 341822160456, 4212806126257, 12980240539227, 159975692596981, 492907318330170, 6074863512559021, 18717497856007233, 230684837784645817

Offset: 0

N. J. A. Sloane

With a(-1) = 0, a(n-1) gives, for n >= 0, the numerator of the convergents to 1/sqrt(40) = 1/(2*sqrt(10)) = A020797. - Wolfdieter Lang, Nov 21 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

Cf. A010494, A041066 (numerators).

I:=[1, 3, 37, 114]; [n le 4 select I[n] else 38*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[40],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011 *)
Denominator[Convergents[Sqrt[40],30]] (* Harvey P. Dale, Sep 12 2013 *)
CoefficientList[Series[-(x^2 - 3 x - 1)/((x^2 - 6 x - 1)(x^2 + 6 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2013 *)

G.f.: -(x^2-3*x-1) / ((x^2-6*x-1)*(x^2+6*x-1)). - Colin Barker, Nov 12 2013

a(n) = 38*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013

More terms from Colin Barker, Nov 12 2013

A337092 Decimal expansion of sqrt(40/Pi).

Original entry on oeis.org

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

Offset: 1

Peter Munn, Aug 15 2020

A gauge point marked c^1 or c_1 ("c" with a superscripted or subscripted "1") on slide rule calculating devices in the 20th century. The Pickworth reference notes its use "in calculating the contents of cylinders".

			3.568248232305...

C. N. Pickworth, The Slide Rule, 24th Ed., Pitman, London, 1945, p. 53, Gauge Points.

Peter Munn, Aristo 89 Slide Rule
Index entries for transcendental numbers

Cf. A002161, A010494, A190732.

evalf(sqrt(40.0/Pi)) ; # R. J. Mathar, Sep 02 2020

RealDigits[Sqrt[40/Pi], 10, 100][[1]] (* Amiram Eldar, Aug 15 2020 *)

sqrt(40/Pi) \\ Michel Marcus, Aug 19 2020

Equals A010494/A002161 = 2*sqrt(10*A049541).

A040033 Continued fraction for sqrt(40).

Original entry on oeis.org

6, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3

Offset: 0

N. J. A. Sloane

			6.3245553203367586639977870... = 6 + 1/(3 + 1/(12 + 1/(3 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 05 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. A010494 (decimal expansion), A040005.

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

ContinuedFraction[Sqrt[40],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
LinearRecurrence[{0,1},{6,3,12},100] (* or *) PadRight[{6},100,{12,3}] (* Harvey P. Dale, Mar 03 2018 *)

{ allocatemem(932245000); default(realprecision, 32000); x=contfrac(sqrt(40)); for (n=0, 20000, write("b040033.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 05 2009

From Stefano Spezia, Jul 27 2025: (Start)
a(n) = 3*A040005(n).
G.f.: 3*(2 + x + 2*x^2)/(1 - x^2). (End)

A171539 Decimal expansion of sqrt(6/35).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

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

			sqrt(6/35) = sqrt(210)/35 = 0.414039335605412530677676118776...

Wikipedia, Table of Clebsch-Gordan coefficients

RealDigits[Sqrt[6/35],10,120][[1]] (* Harvey P. Dale, Jun 09 2023 *)

equals A010464/A010490 = A010494*A171537/10.

cp's OEIS Frontend

A248266 Egyptian fraction representation of sqrt(40) (A010494) using a greedy function.

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A020797 Decimal expansion of 1/sqrt(40).

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A132677 Period 3: repeat [1, 2, -3].

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A041066 Numerators of continued fraction convergents to sqrt(40).

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A041067 Denominators of continued fraction convergents to sqrt(40).

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A337092 Decimal expansion of sqrt(40/Pi).

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A040033 Continued fraction for sqrt(40).

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