A010127 -id:A010127 - 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

A010479 Decimal expansion of square root of 23.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 4 followed by {1, 3, 1, 8} repeated. - Harry J. Smith, Jun 03 2009

			4.7958315233127195415974380641626939199967070419041293464853...

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 275.

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

Cf. A010127 (continued fraction).

RealDigits[N[Sqrt[23], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(23); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010479.txt", n, " ", d));  \\ Harry J. Smith, Jun 03 2009

A307530 Primes p for which the continued fraction expansion of sqrt(p) has a single 1 starting at second position.

Original entry on oeis.org

3, 23, 47, 59, 61, 79, 97, 137, 139, 163, 167, 191, 193, 223, 251, 281, 283, 313, 317, 349, 353, 359, 389, 397, 431, 433, 439, 479, 521, 523, 563, 569, 571, 613, 617, 619, 659, 661, 673, 719, 727, 769, 773, 823, 827, 829, 839, 881, 883, 887, 941, 947, 953, 1009

Offset: 1

Michel Marcus, Apr 13 2019

nonn

Misak and Ulas prove that the density of primes with k ones is 1/(Fibonacci(k+3)*Fibonacci(k+1)) = 1/3, here with k=1 (a single 1).

			For p = 3,  we have [1; 1, 2, ...]; see A040001.
For p = 27, we have [4; 1, 3, ...]; see A010127.
For p = 47, we have [6; 1, 5, ...]; see A010137.

Piotr Miska, Maciej Ulas, On consecutive 1's in continued fractions expansions of square roots of prime numbers, arXiv:1904.03404 [math.NT], 2019. See Corollary 4.3. p. 13.
Index entries for continued fractions for constants

Cf. A040001, A010127, A010137, A307453.

isok(p) = my(cf = contfrac(sqrt(p))); (cf[2] == 1) && (cf[3] != 1);
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", ")));

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A010121 Continued fraction for sqrt(7).

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A010479 Decimal expansion of square root of 23.

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A307530 Primes p for which the continued fraction expansion of sqrt(p) has a single 1 starting at second position.

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PARI