A217562 -id:A217562 - cp's OEIS frontend

A002163 Decimal expansion of square root of 5.

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

Offset: 1

N. J. A. Sloane

Also the limiting ratio of Lucas(n)/Fibonacci(n). - Alexander Adamchuk, Oct 10 2007
Continued fraction expansion is 2 followed by {4} repeated. - Harry J. Smith, Jun 01 2009
This is the first Lagrange number. - Alonso del Arte, Dec 06 2011
Equals Tachiya's Product_{n > 0} (1 + 2/A000032(2^n)) = 4*Product_{n > 0} (1 - 1/A000032(2^n)). - Jonathan Sondow, Jan 11 2012
A computation similar, with that of the universal parabolic constant, performed on the curve cosh(x) with the parameters of the osculating parabola, gives as result 2*sinh(arccosh(3/2)), that is sqrt(5) instead of 2.2955871... for the parabola. - Jean-François Alcover, Jul 18 2013
Because sqrt(5) = -1 + 2*phi, with the golden section phi from A001622, this is an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Jan 08 2018
This constant appears in the theorem of Hurwitz on the best approximation of any irrational number with infinitely many rationals: |theta - h/k| < 1/(sqrt(5)*k^2). See Niven, also for the Hurwitz 1891 reference. - Wolfdieter Lang, May 27 2018
Diameter of a sphere whose surface area equals 5*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Nov 11 2018

			2.236067977499789696409173668731276235440618359611525724270897245410520...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 187, 203.
W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, Theorem 1.5, pp. 6, 14.
Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 106.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. F. Jones, 22900D approximations to the square roots of the primes less than 100, Math. Comp., 22 (1968), 234-235.
Jason Kimberley, Index of expansions of sqrt(d) in base b.
D. Merrill, First million digits of square root of 5. [Broken link]
Robert Nemiroff and Jerry Bonnell, The first 1 million digits of the square root of 5.
Robert Nemiroff and Jerry Bonnell, Plouffe's Inverter, The first 1 million digits of the square root of 5.
Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
Jonathan Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, arXiv:1106.4246 [math.NT], 2011; Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
Y. Tachiya, Transcendence of certain infinite products, J. Number Theory 125 (2007), 182-200.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, J. Int. Seq. 13 (2010) # 10.7.5, eq. (1).
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622.

Cf. A040002 (continued fraction).

SetDefaultRealField(RealField(100)); Sqrt(5); // Vincenzo Librandi, Feb 13 2020

RealDigits[N[Sqrt[5],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

default(realprecision, 20080); x=sqrt(5); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002163.txt", n, " ", d));  \\ Harry J. Smith, Jun 01 2009

e^(i*Pi) + 2*phi = sqrt(5).
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)).
Equals Sum_{n>=0} 25*(2*n+1)!/(n!^2*3^(2*n+3)). (End)
Equals -1 + 2*phi, with phi = A001622. An integer number in the real quadratic number field Q(sqrt(5)). - Wolfdieter Lang, May 09 2018
Equals Sum_{k>=0} binomial(2*k,k)/5^k. - Amiram Eldar, Aug 03 2020
Equals 2*sin(Pi/5) * 2*sin(2*Pi/5). - Gary W. Adamson, Jul 14 2022
Equals w - w^2 - w^3 + w^4 where w = exp(2*Pi*i/5). - Alexander R. Povolotsky, Nov 23 2022
From Antonio Graciá Llorente, Apr 18 2024: (Start)
Equals Product_{k>=0} ((10*k + 2)(10*k + 4)(10*k + 6)(10*k + 8))/((10*k + 1)*(10*k + 3)*(10*k + 7)*(10*k + 9)).
Equals Product_{k>=1} A217562(k)/A045572(k).
Equals Product_{k>=0} (1/2)*(((4*k + 9)/(4*k + 1))^(1/2) + ((4*k + 1)/(4*k + 9))^(1/2)).
Equals Product_{k>=1} (phi^k + phi)/(phi^k + phi - 1), with phi = A001622.
Equals Product_{k>=0} (Fibonacci(2*k + 3) + (-1)^k)/(Fibonacci(2*k + 3) - (-1)^k). (End)

Sequence corrected by Paul Zimmermann, Mar 15 1996

Additional comments from Jason Earls, Mar 26 2001

A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).

Original entry on oeis.org

1, 3, 4, 5, 9, 12, 14, 15, 16, 20, 23, 25, 26, 27, 31, 34, 36, 37, 38, 42, 45, 47, 48, 49, 53, 56, 58, 59, 60, 64, 67, 69, 70, 71, 75, 78, 80, 81, 82, 86, 89, 91, 92, 93, 97, 100, 102, 103, 104, 108, 111, 113, 114, 115, 119, 122, 124, 125, 126, 130, 133, 135, 136, 137

Offset: 0

Bruno Berselli, Jan 20 2016

(m^k-1)/11 is a nonnegative integer when
. m is a member of this sequence and k is an odd multiple of 5 (A017329),
. m is a member of A017401 and k is odd but not multiple of 5 (A045572),
. m is a member of A175885 and k is even but not multiple of 5 (A217562),
. m is a member of A160542 and k is a positive multiple of 10 (A008592),
apart from the trivial case in which k=0.
Also, numbers that are congruent to {1, 3, 4, 5, 9} mod 11. Therefore, the product of two terms belongs to the sequence.
Union of this sequence and A267541 is A160542.
a(n) is prime for n = 1, 3, 10, 14, 17, 21, 24, 27, 30, 33, 40, 44, 47, ...

			From the linear recurrence:
(-A267541) ..., -13, -10, -8, -7, -6, -2, 1, 3, 4, 5, 9, 12, ... (A267755)

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A160542, A017329, A267541.

Related sequences (see the first comment): A017401, A160542, A175885.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)));

I:=[1,3,4,5,9,12]; [n le 6 select I[n] else Self(n-1)+Self(n-5)-Self(n-6): n in [1..70]]; // Vincenzo Librandi, Jan 21 2016

gf := (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6): deg := 64: series(gf, x, deg): seq(coeff(%, x, n), n=0..deg-1); # Peter Luschny, Jan 21 2016

CoefficientList[Series[(1 + 2 x + x^2 + x^3 + 4 x^4 + 2 x^5)/(1 - x - x^5 + x^6), {x, 0, 70}], x]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {1, 3, 4, 5, 9, 12}, 70]
Select[Range[140], MemberQ[{1, 3, 4, 5, 9}, Mod[#, 11]]&]

Vec((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)+O(x^70))

gf = (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6)
print(taylor(gf, x, 0, 63).list()) # Peter Luschny, Jan 21 2016

G.f.: (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(-n) = -A267541(n-1).
a(n) = n + 1 + 2*floor(n/5) + 3*floor((n+1)/5) + floor((n+4)/5). - Ridouane Oudra, Sep 06 2023

A361913 a(n) is the number of steps in the main loop of the Pollard rho integer factorization algorithm for n, with x=2, y=2 and g(x)=x^2-1.

Original entry on oeis.org

2, 2, 2, 1, 2, 4, 2, 2, 1, 2, 2, 3, 2, 1, 2, 4, 2, 3, 1, 2, 2, 8, 2, 1, 2, 2, 2, 4, 1, 4, 2, 2, 2, 1, 2, 6, 2, 2, 1, 5, 2, 6, 2, 1, 2, 11, 2, 4, 1, 2, 2, 5, 2, 1, 2, 2, 2, 7, 1, 3, 2, 2, 2, 1, 2, 4, 2, 2, 1, 3, 2, 8, 2, 1, 2, 2, 2, 10, 1, 2, 2, 12, 2, 1, 2, 2

Offset: 2

Darío Clavijo, Mar 29 2023

nonn

x=2 and y=2 are the minimum effective values for Pollard rho, but any x = y > 2 would give the same answer.
n is in A217562 if gcd(n, a(n)) > 1.
n is in A047201 if gcd(phi(n), a(n)) > 1, where phi is Euler's totient function.

Eric Weisstein's World of Mathematics, Pollard rho Factorization Method.
Wikipedia, Pollard's rho algorithm

Cf. A005563.

from gmpy2 import *
def a(n):
  c,d,x,y,g = 0,1,2,2,lambda x:pow(x,2,n)-1
  while d == 1:
    c,x,y =c+1,g(x),g(g(y))
    d = gcd(abs(x-y), n)
  return c

cp's OEIS Frontend

A383227 a(n) is the product of first n even numbers not divisible by 5 (cf. A217562).

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A002163 Decimal expansion of square root of 5.

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A267755 Expansion of (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6).

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A361913 a(n) is the number of steps in the main loop of the Pollard rho integer factorization algorithm for n, with x=2, y=2 and g(x)=x^2-1.

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A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).