A010123 -id:A010123 - cp's OEIS frontend

A010471 Decimal expansion of square root of 14.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 3 followed by {1, 2, 1, 6} repeated. - Harry J. Smith, Jun 02 2009

The convergents are given in A041020/A041021. - Wolfdieter Lang, Nov 27 2017

			3.741657386773941385583748732316549301756019807778726946303745467320035...

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

Cf. A010123 (continued fraction), A041020/A041021.

evalf[100](sqrt(14)); # Muniru A Asiru, Feb 12 2019

RealDigits[N[Sqrt[14], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)
RealDigits[Sqrt[14],10,120][[1]] (* Harvey P. Dale, Jan 07 2023 *)

default(realprecision, 20080); x=sqrt(14); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010471.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

A010121 Continued fraction for sqrt(7).

Original entry on oeis.org

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

A010696 Periodic sequence: Repeat 2,6.

Original entry on oeis.org

2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2

Offset: 0

N. J. A. Sloane

Original name: Period 2.

Also continued fraction expansion of 1+(2/3)*sqrt(3). - Bruno Berselli, Sep 22 2011

Index entries for linear recurrences with constant coefficients, signature (0,1)

Cf. A174114. [From Reinhard Zumkeller, Mar 08 2010]

PadRight[{}, 100, {2, 6}] (* Paolo Xausa, Feb 22 2024 *)

G.f.: ( -2-6*x ) / ( (x-1)*(1+x) ). - R. J. Mathar, Jul 07 2011

a(n) = a(-n) = 2*A010684(n) = A131800(2n+1) = A010123(2n+2). - Bruno Berselli, Sep 22 2011

Definition rewritten by Bruno Berselli, Sep 22 2011

A326422 Numbers k such that A000045(k) mod 5 is prime.

Original entry on oeis.org

3, 4, 6, 7, 13, 14, 16, 17, 23, 24, 26, 27, 33, 34, 36, 37, 43, 44, 46, 47, 53, 54, 56, 57, 63, 64, 66, 67, 73, 74, 76, 77, 83, 84, 86, 87, 93, 94, 96, 97, 103, 104, 106, 107, 113, 114, 116, 117, 123, 124, 126, 127, 133, 134, 136, 137, 143, 144, 146, 147, 153, 154, 156, 157

Offset: 1

Vincenzo Librandi, Jul 06 2019

Position of prime numbers in A082116.

Vincenzo Librandi, Table of n, a(n) for n = 1..3740
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A000045, A082116.

Partial sums of A010123.

[n: n in [0..200] | IsPrime(Fibonacci(n) mod 5)];

Select[Range[160], MemberQ[{2, 3}, Mod[Fibonacci[#], 5]] &]

G.f.: x*(3*x^2+4*x+3)*(x^2-x+1)/((x+1)*(x^2+1)*(x-1)^2). - Alois P. Heinz, Jul 08 2019

E.g.f.: (6 - 2*cos(x) + (5*x - 4)*cosh(x) + 2*sin(x) + (5*x - 1)*sinh(x))/2. - Stefano Spezia, Jul 15 2025

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A010471 Decimal expansion of square root of 14.

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A010121 Continued fraction for sqrt(7).

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A010696 Periodic sequence: Repeat 2,6.

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A326422 Numbers k such that A000045(k) mod 5 is prime.

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