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%I #22 Jul 21 2023 13:48:15
%S 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,
%T 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,
%U 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
%N Decimal expansion of (3+sqrt(21))/6.
%C Continued fraction expansion of (3+sqrt(21))/6 is A010684.
%C 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
%H Daniel Starodubtsev, <a href="/A176014/b176014.txt">Table of n, a(n) for n = 1..10000</a>
%H Hugo Pfoertner, <a href="/A005186/a005186.png">Ratio of successive terms in A005186</a>, illustration of deviation from (3+sqrt(21))/6.
%e (3+sqrt(21))/6 = 1.26376261582597333443...
%t RealDigits[(3+Sqrt[21])/6,10,120][[1]] (* _Harvey P. Dale_, Jul 21 2023 *)
%o (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
%Y Cf. A010477 (decimal expansion of sqrt(21)).
%Y Cf. A010684 (repeat 1, 3), A136210, A136211.
%Y Cf. A005186, A006577, A127824, A176866.
%K cons,nonn
%O 1,2
%A _Klaus Brockhaus_, Apr 06 2010
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