A010170 -id:A010170 - cp's OEIS frontend

A010550 Decimal expansion of square root of 99.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 9 followed by {1, 18} repeated. - Harry J. Smith, Jun 12 2009

			9.949874371066199547344798210012060051781265636768060791176046438349453...

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

Cf. A010170 Continued fraction.

RealDigits[N[99^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 24 2012 *)

{ default(realprecision, 20080); x=sqrt(99); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010550.txt", n, " ", d)); } \\ Harry J. Smith, Jun 12 2009

A041179 Denominators of continued fraction convergents to sqrt(99).

Original entry on oeis.org

1, 1, 19, 20, 379, 399, 7561, 7960, 150841, 158801, 3009259, 3168060, 60034339, 63202399, 1197677521, 1260879920, 23893516081, 25154396001, 476672644099, 501827040100, 9509559365899, 10011386405999, 189714514673881, 199725901079880, 3784780734111721

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 18 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,20,0,-1).

Cf. A041178, A010170, A020856, A010550, A002530.

I:=[1, 1, 19, 20]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013

Denominator[Convergents[Sqrt[99], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

G.f.: -(x^2-x-1) / (x^4-20*x^2+1). - Colin Barker, Nov 14 2013
a(n) = 20*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 12 2013
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(18) + sqrt(22) )/2 and beta = ( sqrt(18) - sqrt(22) )/2 be the roots of the equation x^2 - sqrt(18)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 18 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 18*a(2*n) + a(2*n - 1). (End)

More terms from Colin Barker, Nov 14 2013

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A010550 Decimal expansion of square root of 99.

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