A176014 Decimal expansion of (3+sqrt(21))/6.

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

Offset

1,2

Comments

Continued fraction expansion of (3+sqrt(21))/6 is A010684.

Also greatest eigenvalue of the 6 X 6 matrix [[3 0 0 3 0 0][0 0 0 0 1 0][0 3 0 0 3 0][0 0 0 0 1 0][0 0 3 0 0 3][0 0 0 0 1 0]]/3. It is conjectured that this is lim_{k->infinity} A005186(k+1)/A005186(k), i.e., the asymptotic growth rate of the number of numbers with the same total stopping time in the Collatz iteration. - Hugo Pfoertner, Sep 28 2020

Merlini and Sala (1999) give the value as (1 + sqrt(7/3))/2 and call it "Collatz's constant". - M. F. Hasler, Nov 18 2025

References

Danilo Merlini and Nicoletta Sala, On the Fibonacci's Attractor and the Long Orbits in the 3n+1 Problem, International Journal of Chaos Theory and Applications, Vol. 4, No. 2-3 (1999), 75-84.

Links

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Hugo Pfoertner, Ratio of successive terms in A005186, illustration of deviation from (3+sqrt(21))/6.

Index entries for algebraic numbers, degree 2.

Formula

Equals (1 + sqrt(7/3))/2. - M. F. Hasler, Nov 18 2025

Minimal polynomial: 3*x^2 - 3*x - 1. - Amiram Eldar, May 05 2026

Example

1.26376261582597333443134119895466808149740942946132...

Mathematica

RealDigits[(3+Sqrt[21])/6, 10, 120][[1]] (* Harvey P. Dale, Jul 21 2023 *)

Prog

(PARI) vecmax(mateigen([1, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1/3, 0; 0, 1, 0, 0, 1, 0; 0, 0, 0, 0, 1/3, 0; 0, 0, 1, 0, 0, 1; 0, 0, 0, 0, 1/3, 0], 1)[1]) \\ Hugo Pfoertner, Sep 28 2020

Crossrefs

Cf. A010477 (decimal expansion of sqrt(21)).

Cf. A010684 (repeat 1, 3), A136210, A136211.

Cf. A005186, A006577, A127824, A176866, A391769.

Sequence in context: A354372 A354111 A154129 * A011447 A376468 A076041

Adjacent sequences: A176011 A176012 A176013 * A176015 A176016 A176017

Keyword

cons,nonn

Author

Klaus Brockhaus, Apr 06 2010

Status

approved