A040015 -id:A040015 - cp's OEIS frontend

A010476 Decimal expansion of square root of 20.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 4 followed by {2, 8} repeated. - Harry J. Smith, Jun 03 2009

			4.472135954999579392818347337462552470881236719223051448541794490821041....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Except for offset, same as A020762.
Cf. A040015 (continued fraction). - Harry J. Smith, Jun 03 2009
Cf. A002163 (decimal expansion of square root of 5).

RealDigits[N[Sqrt[20], 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=sqrt(20); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010476.txt", n, " ", d));  \\ Harry J. Smith, Jun 03 2009

sqrt(20) = 2*sqrt(5). - Alonso del Arte, Jun 26 2015

Equals Sum_{k>=0} binomial(2*k,k) * k/5^k. - Amiram Eldar, Aug 03 2020

A041031 Denominators of continued fraction convergents to sqrt(20).

Original entry on oeis.org

1, 2, 17, 36, 305, 646, 5473, 11592, 98209, 208010, 1762289, 3732588, 31622993, 66978574, 567451585, 1201881744, 10182505537, 21566892818, 182717648081, 387002188980, 3278735159921, 6944472508822

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).

Cf. A010476, A040015, A041030 (numerators).

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[20],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
a0[n_] := ((5+2*Sqrt[5])/(9+4*Sqrt[5])^n+(5-2*Sqrt[5])*(9+4*Sqrt[5])^n)/10 //Simplify
a1[n_] := (-1/(9+4*Sqrt[5])^n+(9+4*Sqrt[5])^n)/(4*Sqrt[5]) //Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)

G.f.: (1+2*x-x^2)/(1-18*x^2+x^4). - Colin Barker, Jan 01 2012
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((5+2*sqrt(5))/(9+4*sqrt(5))^n+(5-2*sqrt(5))*(9+4*sqrt(5))^n)/10.
a1(n) = (-1/(9+4*sqrt(5))^n+(9+4*sqrt(5))^n)/(4*sqrt(5)). (End)

A040185 Continued fraction for sqrt(200).

Original entry on oeis.org

14, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28, 7, 28

Offset: 0

N. J. A. Sloane

Cf. A040000 (for sqrt(2)), A040015 (for sqrt(20))

with(numtheory): Digits := 300: convert(evalf(sqrt(200)),confrac);

ContinuedFraction[Sqrt[200],300] (* Vladimir Joseph Stephan Orlovsky, Mar 28 2011 *)
PadRight[{14},120,{28,7}] (* Harvey P. Dale, Feb 08 2025 *)

A064850 Period of continued fraction for sqrt(5)*n.

Original entry on oeis.org

1, 2, 6, 2, 5, 4, 10, 4, 2, 14, 12, 4, 5, 10, 28, 8, 1, 2, 4, 14, 6, 8, 6, 4, 31, 14, 10, 12, 12, 20, 8, 20, 20, 2, 52, 2, 19, 4, 28, 24, 18, 8, 50, 12, 28, 6, 10, 4, 70, 62, 8, 18, 7, 10, 6, 8, 8, 12, 72, 20, 3, 12, 8, 36, 41, 28, 86, 2, 6, 44, 84, 2, 43, 42, 120, 4, 52, 36, 28, 44, 38

Offset: 1

R. K. Guy, Oct 26 2001

nonn

			A040002 (cfrac for n=1) has period length 1, so a(1)=1. A040015 (cfrac for n=2) has period length 2, so a(2)=2. A010135 (cfrac for n=3) has period length 6, so a(3)=6. - _R. J. Mathar_, Feb 10 2016

Table[Length[ContinuedFraction[Sqrt[5]n][[2]]],{n,90}] (* Harvey P. Dale, Apr 13 2015 *)

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A010476 Decimal expansion of square root of 20.

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A041031 Denominators of continued fraction convergents to sqrt(20).

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A040185 Continued fraction for sqrt(200).

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A064850 Period of continued fraction for sqrt(5)*n.

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