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

Author

Klaus Brockhaus, Apr 06 2010

Keywords

• cons
• nonn

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

Examples

			(3+sqrt(21))/6 = 1.26376261582597333443...
		

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.

Crossrefs

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

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

Cf. A005186, A006577, A127824, A176866.

Programs

Mathematica

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

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