A278809 - OEIS

%I #18 Dec 11 2016 08:42:41

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

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

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

%N Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278449, where c=3 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

%C For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.

%C If b(1) were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.

%C The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=3 similar): "Suppose one starts with b(1) = 2, the sequence b(n) would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."

%H Rok Cestnik, <a href="/A278809/b278809.txt">Table of n, a(n) for n = 1..1000</a>

%H Rok Cestnik, <a href="/A278809/a278809.pdf">Plot of the dependence of b(1) on c</a>

%F log_3(2*log_3(3*log_3(4*log_3(...)))). - _Andrey Zabolotskiy_, Dec 01 2016

%e 1.08287360952073869408285031345310080257863454785385...

%t c = 3;

%t n = 100;

%t acc = Round[n*1.2];

%t th = 1000000;

%t b1 = 0;

%t For[p = 0, p < acc, ++p,

%t   For[d = 0, d < 9, ++d,

%t     b1 = b1 + 1/10^p;

%t     bn = b1;

%t     For[i = 1, i < Round[n*1.2], ++i,

%t      bn = N[c^(bn/i), acc];

%t      If[bn > th, Break[]];

%t      ];

%t     If[bn > th, {

%t       b1 = b1 - 1/10^p;

%t       Break[];

%t       }];

%t     ];

%t   ];

%t N[b1,n]

%t RealDigits[ Fold[ Log[3, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* _Robert G. Wilson v_, Dec 02 2016 *)

%Y For sequence round(b(n)) see A278449.

%Y For different values of c see A278808, A278810, A278811, A278812.

%Y For b(1)=0 see A278813.

%K nonn,cons

%O 1,3

%A _Rok Cestnik_, Nov 28 2016