A010148 -id:A010148 - cp's OEIS frontend

A010521 Decimal expansion of square root of 69.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 8 followed by {3, 3, 1, 4, 1, 3, 3, 16} repeated. - Harry J. Smith, Jun 08 2009

			8.306623862918074852584262744907492010232214248955655779432188369037585...

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

Cf. A010148 Continued fraction.

RealDigits[N[69^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2012 *)

{ default(realprecision, 20080); x=sqrt(69); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010521.txt", n, " ", d)); } \\ Harry J. Smith, Jun 08 2009

A067280 Number of terms in continued fraction for sqrt(n), excl. 2nd and higher periods.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 5, 3, 1, 2, 3, 3, 6, 5, 3, 1, 2, 3, 7, 3, 7, 7, 5, 3, 1, 2, 3, 5, 6, 3, 9, 5, 5, 5, 3, 1, 2, 3, 3, 3, 4, 3, 11, 9, 7, 13, 5, 3, 1, 2, 3, 7, 6, 7, 5, 3, 7, 8, 7, 5, 12, 5, 3, 1, 2, 3, 11, 3, 9, 7, 9, 3, 8, 6, 5, 13, 7, 5, 5, 3, 1, 2, 3, 3, 6, 11, 3, 7, 6, 3, 9, 9, 11, 17, 5, 5, 12, 5

Offset: 1

Frank Ellermann, Feb 23 2002

			a(2)=2: [1,(2)+ ]; a(3)=3: [1,(1,2)+ ]; a(4)=1: [2]; a(5)=2: [2,(4)+ ].

H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th edition, 1999, table 1.

Related sequences: 2 : A040000, ..., 44: A040037, 48: A040041, ..., 51: A040043, 56: A040048, 60: A040052, 63: A040055, ..., 66: A040057. 68: A040059, 72: A040063, 80: A040071.
Related sequences: 45: A010135, ..., 47: A010137, 52: A010138, ..., 55: A010141, 57: A010142, ..., 59: A010144. 61: A010145, 62: A010146. 67: A010147, 69: A010148, ..., 71: A010150.
Cf. A003285.

from sympy import continued_fraction_periodic
def A067280(n): return len((a := continued_fraction_periodic(0,1,n))[:1]+(a[1] if a[1:] else [])) # Chai Wah Wu, Jun 14 2022

a(n) = A003285(n) + 1. - Andrey Zabolotskiy, Jun 23 2020

Name clarified by Michel Marcus, Jun 22 2020

A041121 Denominators of continued fraction convergents to sqrt(69).

Original entry on oeis.org

1, 3, 10, 13, 62, 75, 287, 936, 15263, 46725, 155438, 202163, 964090, 1166253, 4462849, 14554800, 237339649, 726573747, 2417060890, 3143634637, 14991599438, 18135234075, 69397301663, 226327139064, 3690631526687, 11298221719125, 37585296684062, 48883518403187

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 (0,0,0,0,0,0,0,15550,0,0,0,0,0,0,0,-1).

Cf. A041120, A010148, A020826, A010521.

I:=[1, 3, 10, 13, 62, 75, 287, 936, 15263, 46725, 155438, 202163, 964090, 1166253, 4462849, 14554800]; [n le 16 select I[n] else 15550*Self(n-8)-Self(n-16): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[69],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Denominator[Convergents[Sqrt[69], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,15550,0,0,0,0,0,0,0,-1},{1,3,10,13,62,75,287,936,15263,46725,155438,202163,964090,1166253,4462849,14554800},30] (* Harvey P. Dale, Oct 18 2015 *)

G.f.: -(x^14 -3*x^13 +10*x^12 -13*x^11 +62*x^10 -75*x^9 +287*x^8 -936*x^7 -287*x^6 -75*x^5 -62*x^4 -13*x^3 -10*x^2 -3*x -1) / (x^16 -15550*x^8 +1). - Colin Barker, Nov 13 2013

a(n) = 15550*a(n-8) - a(n-16). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 13 2013

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A010521 Decimal expansion of square root of 69.

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A067280 Number of terms in continued fraction for sqrt(n), excl. 2nd and higher periods.

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A041121 Denominators of continued fraction convergents to sqrt(69).

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