A178255 -id:A178255 - cp's OEIS frontend

A109007 a(n) = gcd(n,3).

3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1

Offset: 0

Mitch Harris

For n>1: a(n) = GCD of the n-th and (n+2)-th triangular numbers = A050873(A000217(n+2), A000217(n)). - Reinhard Zumkeller, May 28 2007
From Klaus Brockhaus, May 24 2010: (Start)
Continued fraction expansion of (3+sqrt(17))/2.
Decimal expansion of 311/999. (End)

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

Cf. A000217, A026741, A050873, A109004, A130334.

Cf. A178255 (decimal expansion of (3+sqrt(17))/2). - Klaus Brockhaus, May 24 2010

[Gcd(n,3) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2016

A109007:=n->gcd(n,3): seq(A109007(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2016

GCD[Range[0,100],3] (* or *) PadRight[{},110,{3,1,1}] (* Harvey P. Dale, Jun 28 2015 *)

a(n)=gcd(n,3) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 1 + 2*[3|n] = 1 + 2(1 + 2*cos(2*n*Pi/3))/3, where [x|y] = 1 when x divides y, 0 otherwise.
a(n) = a(n-3) for n>2.
Multiplicative with a(p^e, 3) = gcd(p^e, 3). - David W. Wilson, Jun 12 2005
O.g.f.: -(3+x+x^2)/((x-1)*(x^2+x+1)). - R. J. Mathar, Nov 24 2007
Dirichlet g.f. zeta(s)*(1+2/3^s). - R. J. Mathar, Apr 08 2011
a(n) = 2*floor(((n-1) mod 3)/2) + 1. - Gary Detlefs, Dec 28 2011
a(n) = 3^(1 - sgn(n mod 3)). - Wesley Ivan Hurt, Jul 24 2016
a(n) = 3/(1 + 2*((n^2) mod 3)). - Timothy Hopper, Feb 25 2017
a(n) = (5 + 4*cos(2*n*Pi/3))/3. - Wesley Ivan Hurt, Oct 04 2018

A222132 Decimal expansion of sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))).

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 08 2013

Sequence with a(1) = 1 is decimal expansion of sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))) = A222133.
Because 17 == 1 (mod 4), the basis for integers in the real quadratic number field K(sqrt(17)) is <1, omega(17)>, where omega(17) = (1 + sqrt(17))/2. - Wolfdieter Lang, Feb 10 2020
This is the positive root of the polynomial x^2 - x - 4, with negative root -A222133. - Wolfdieter Lang, Dec 10 2022
It is the spectral radius of the diamond graph (see Seeger and Sossa, 2023). - Stefano Spezia, Sep 19 2023
c^n = A006131(n) + A006131(n-1) * d, where c = (1 + sqrt(17))/2 and d = (-1 + sqrt(17))/2. - Gary W. Adamson, Nov 25 2023
c^n = A052923(n) + A006131(n-1) * c. Also for negative n. - Wolfdieter Lang, Nov 27 2023
The effective degree of maximal entropy random walk on the barred-square graph (see Burda et al.). - Stefano Spezia, Feb 07 2025

			2.561552812808830274910704...

J. Burda, J. Duda, J. M. Luck, and B. Waclaw, The Various Facets of Random Walk Entropy, Acta Phys. Pol. B 41, 949-987 (2010). See pp. 964, 969.
Alberto Seeger and David Sossa, Infinite families of connected graphs with equal spectral radius, Australas. J. Combin. 87 (2) (2023), 258-276. See pp. 260 and 263.

Cf. A222133, A178255, A082486.

Cf. A006131, A052923.

Digits:=140:
evalf((sqrt(17)+1)/2);  # Alois P. Heinz, Sep 19 2023

Mathematica
```
RealDigits[(1 + Sqrt[17])/2, 10, 130]
```

Closed form: (sqrt(17) + 1)/2 = A178255 - 1 = A082486 - 2.

sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))) - 1 = sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))). See A222133.

A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 06 2003

This constant can also be defined as the root of coth x = x, as this equation and the above are equivalent. - Carl R. White, Dec 09 2003. Also the root of x*tanh x = 1. - N. J. A. Sloane, May 07 2020
This constant is also the point on the parametric tractrix (t - tanh(t), sech(t)) the least distant from the origin. - Michael Clausen, Feb 18 2013
This constant also equals sqrt(lambda^2+1), where lambda is the Laplace limit constant A033259. - Jean-François Alcover, Sep 08 2014, after Steven Finch.
For each of the real symmetric n X n matrices M defined by M(i,j) = max(i,j) with n >= 2, there exist n-1 negative eigenvalues < -1/4 and only one positive eigenvalue lambda(n) such that n^2/2 < lambda(n) < n^2. Indeed, when n tends to infinity, lambda(n) ~ n^2/(this constant)^2 (see reference O. Carton et al.). For n = 2, the positive eigenvalue is (3+sqrt(17))/2 [A178255]. - Bernard Schott, Mar 13 2020

			1.1996786402577338339163698486411419442614587884186072...

O. Carton, L. Rosaz, M. Zeitoun, Problèmes corrigés de Mathématiques posés au Concours de Mines/Ponts, Tome 5, Ellipses, 1992; Problème Mines-Ponts 1991 - Options M, P', TA - Epreuve pratique p. 125.
Steven R. Finch, Mathematical constants, Volume 94, Encyclopedia of mathematics and its applications, Cambridge University Press, 2003, p. 268.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:9 at page 283.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jogundas Armaitis, Molecules and Polarons in Extremely Imbalanced Fermi Mixtures, Master's Thesis, Aug 11 2011, Institute for Theoretical Physics, Utrecht University.
Robert Ferréol, Tractrix, Mathcurve.
D. E. Knuth, Whirlpool Permutations, May 05 2020.
Mathematics Stack Exchange, Laplace limit constant.
Eric Weisstein's World of Mathematics, Kepler's Equation.
Eric Weisstein's World of Mathematics, Laplace Limit.
Eric Weisstein's World of Mathematics, Hyperbolic Cotangent
Wikipedia, Tractrix.
Index entries for transcendental numbers.

Cf. A003957 (x = cos(x)), A009379, A033259, A069855, A209289.

RealDigits[ x /. FindRoot[ Coth[x] == x, {x, 1}, WorkingPrecision -> 102]] // First (* Jean-François Alcover, Feb 08 2013 *)
1+2 NSum[LaguerreL[n-1,1,4 n]/n Exp[-2 n],{n,1,Infinity}] //
  (* Aaron Hendrickson, Mar 17 2021 *)

solve(u=1,2,tanh(u)-1/u)  /* type e.g. \p99 to get 99 digits; M. F. Hasler, Feb 01 2011 */

Equals 1 + 2*Sum_{n>=1} (Laguerre(n-1,1,4n)/n)*e^(-2n) (see Mathematics Stack Exchange in Links). - Aaron Hendrickson, Mar 17 2022

A190264 Decimal expansion of (sqrt(89) - 6)/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 07 2011

The rectangle R whose shape (i.e., length/width) is (-6 + sqrt(89))/2 can be partitioned into rectangles of shapes 3/2 and 3 in a manner that matches the periodic continued fraction [3/2, 3, 3/2, 3, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [1, 1, 2, 1, 1, 6, 1, 36, 1, 6, 1, 1, 2, 1, 8, 1, 2, 1, 1, 6, 1, 36, ...]. For details, see A188635.

Quadratic number with denominator 2 and minimal polynomial 4x^2 + 24x - 53. - Charles R Greathouse IV, Apr 21 2016

			1.716990566028301905660330188811320358491...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A188635, A190290, A178255.

FromContinuedFraction[{3/2, 3, {3/2, 3}}]
ContinuedFraction[%, 100]  (* [1, 1, 2, 1, 1, 6, 1, 36, ... *)
RealDigits[N[%%, 120]]     (* A190264 *)
N[%%%, 40]

sqrt(89)/2-3 \\ Charles R Greathouse IV, Apr 21 2016

A222133 Decimal expansion of sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))).

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 08 2013

Sequence with a(1) = 2 is the decimal expansion of sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))) = - A222132.

This is the positive root of the minimal polynomial x^2 + x - 4, with negative root -A222132. - Wolfdieter Lang, Dec 10 2022

			1.561552812808830274910704...

Cf. A178255, A082486, A222132.

Mathematica
```
RealDigits[(Sqrt[17] - 1)/2, 10, 130]
```

Closed form: (sqrt(17) - 1)/2 = A178255-2 = A082486-3 = A222132-1.

sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))) + 1 = sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))). See A222132.

cp's OEIS Frontend

A109007 a(n) = gcd(n,3).

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A222132 Decimal expansion of sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))).

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A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

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A190264 Decimal expansion of (sqrt(89) - 6)/2.

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Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A222133 Decimal expansion of sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula