A040043 -id:A040043 - cp's OEIS frontend

A010504 Decimal expansion of square root of 51.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 7 followed by {7, 14} repeated. - Harry J. Smith, Jun 06 2009

			7.141428428542849997999399811367265278766171159902733833208430882765820...

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

Cf. A040043 (continued fraction).

Digits:=100; evalf(sqrt(51)); # Wesley Ivan Hurt, Mar 04 2014

RealDigits[N[Sqrt[51],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)

{ default(realprecision, 20080); x=sqrt(51); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010504.txt", n, " ", d)); } \\ Harry J. Smith, Jun 06 2009

Equals A002194*A010473. - R. J. Mathar, Jun 08 2025

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

A041087 Denominators of continued fraction convergents to sqrt(51).

Original entry on oeis.org

1, 7, 99, 700, 9899, 69993, 989801, 6998600, 98970201, 699790007, 9896030299, 69972002100, 989504059699, 6996500419993, 98940509939601, 699580069997200, 9893061489900401, 69951010499300007, 989207208480100499, 6994401469860003500, 98910827786520149499

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 (0,100,0,-1).

Cf. A041086, A040043, A020808.

I:=[1, 7, 99, 700]; [n le 4 select I[n] else 100*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[51], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
CoefficientList[Series[(1 + 7 x - x^2)/(x^4 - 100 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

G.f.: -(x^2-7*x-1) / (x^4-100*x^2+1). - Colin Barker, Nov 12 2013

a(n) = 100*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 12 2013

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A010504 Decimal expansion of square root of 51.

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

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A041087 Denominators of continued fraction convergents to sqrt(51).

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