A010146 -id:A010146 - cp's OEIS frontend

A010121 Continued fraction for sqrt(7).

2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4

Offset: 0

N. J. A. Sloane

This is a basic member of a family of 4-periodic multiplicative sequences with two parameters (c1,c2), defined for n >= 1 by a(n)=1 if n is odd, a(n)=c1 if n == 0 (mod 4) and a(n)=c2 if n == 2 (mod 4). Here, (c1,c2)=(4,1).
The Dirichlet generating function is (1+(c2-1)/2^s+(c1-c2)/4^s)*zeta(s).
Other members are A010123 with parameters (6,2), A010127 (8,3), A010130 (10,1), A010131 (10,2), A010132 (10,4), A010137 (12,5), A010146 (14,6), A089146 (4,8), A109008 (4,2), A112132 (7,3). If c1=c2, this reduces to the cases discussed in A040001. - R. J. Mathar, Feb 18 2011

			2.645751311064590590501615753...  = A010465 = 2 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...)))).

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
C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq. 14 (2011) # 11.1.4, example 5.
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010465 (decimal expansion).

ContinuedFraction[Sqrt[7],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
CoefficientList[Series[(2 x^2 + 3 x + 2) (x^2 - x + 1) / ((1 - x) (1 + x) (x^2 + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 26 2016 *)
PadRight[{2},120,{4,1,1,1}] (* Harvey P. Dale, Nov 30 2019 *)

{ allocatemem(932245000); default(realprecision, 13000); x=contfrac(sqrt(7)); for (n=0, 20000, write("b010121.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

From R. J. Mathar, Jun 17 2009: (Start)
G.f.: -(2*x^2+3*x+2)*(x^2-x+1)/((x-1)*(1+x)*(x^2+1)).
a(n) = a(n-4), n > 4. (End)
a(n) = (7 + 3*(-1)^n + 3*(-i)^n + 3*i^n)/4, n > 0, where i is the imaginary unit. - Bruno Berselli, Feb 18 2011

A010515 Decimal expansion of square root of 62.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Sqrt(62) = 787400 * Sum_{n>=0} (A001790(n)/2^A005187(floor(n/2)) * 10^(-6n-5)) where A001790(n) are numerators in expansion of 1/sqrt(1-x) and the denominators in expansion of 1/sqrt(1-x) are 2^A005187(n). 786400 = 62*12700, see A020819 (expansion of 1/sqrt(62)). - Gerald McGarvey, Jan 01 2005

Continued fraction expansion is 7 followed by {1, 6, 1, 14} repeated. - Harry J. Smith, Jun 07 2009

			7.874007874011811019685034448812007863681086122020853794598842550313760...

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

Cf. A001790, A005187, A020819.

Cf. A010146 Continued fraction.

RealDigits[N[62^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2012 *)

{ default(realprecision, 20080); x=sqrt(62); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010515.txt", n, " ", d)); } \\ Harry J. Smith, Jun 07 2009

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 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

A041109 Denominators of continued fraction convergents to sqrt(62).

Original entry on oeis.org

1, 1, 7, 8, 119, 127, 881, 1008, 14993, 16001, 110999, 127000, 1888999, 2015999, 13984993, 16000992, 237998881, 253999873, 1761998119, 2015997992, 29985970007, 32001967999, 221997778001, 253999746000, 3777994222001, 4031993968001, 27969958030007

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 (0,0,0,126,0,0,0,-1).

Cf. A041108, A010146, A020819, A010515.

I:=[1, 1, 7, 8, 119, 127, 881, 1008]; [n le 8 select I[n] else 126*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Denominator[Convergents[Sqrt[62], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,0,0,126,0,0,0,-1},{1,1,7,8,119,127,881,1008},30] (* Harvey P. Dale, Oct 26 2016 *)

G.f.: -(x^2-x-1)*(x^4+8*x^2+1) / (x^8-126*x^4+1). - Colin Barker, Nov 12 2013

a(n) = 126*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 12 2013

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A010121 Continued fraction for sqrt(7).

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A010515 Decimal expansion of square root of 62.

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

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A041109 Denominators of continued fraction convergents to sqrt(62).

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