cp's OEIS Frontend

A278811 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278451, where c=5 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

%I A278811 #15 Dec 03 2016 12:25:29
%S A278811 0,1,7,7,5,8,1,9,1,8,8,0,2,5,1,4,0,3,3,3,8,3,5,0,3,1,8,1,3,0,8,6,6,9,
%T A278811 8,5,7,8,8,3,2,9,7,7,0,3,4,6,8,1,0,5,2,1,5,6,4,2,3,6,3,5,7,4,3,3,3,1,
%U A278811 7,4,8,3,6,8,4,2,2,1,1,8,3,5,1,4,8,4,6,9,0,7,6,9,7,1,4,2,7,2,6,5,7,5,1,5,6,9,2,7,7,0,1,6,5,4,1,3,4,9,9,8,6,1,3,5,5,3,1,5,8,5
%N A278811 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278451, where c=5 and b(1) is chosen such that the sequence neither explodes nor goes to 1.
%C A278811 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 A278811 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 A278811 The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=5 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 A278811 Rok Cestnik, <a href="/A278811/b278811.txt">Table of n, a(n) for n = 1..1000</a>
%H A278811 Rok Cestnik, <a href="/A278811/a278811.pdf">Plot of the dependence of b(1) on c</a>
%F A278811 log5(2*log5(3*log5(4*log5(...)))). - _Andrey Zabolotskiy_, Nov 30 2016
%e A278811 0.17758191880251403338350318130866985788329770346810...
%t A278811 c = 5;
%t A278811 n = 100;
%t A278811 acc = Round[n*1.2];
%t A278811 th = 1000000;
%t A278811 b1 = 0;
%t A278811 For[p = 0, p < acc, ++p,
%t A278811   For[d = 0, d < 9, ++d,
%t A278811     b1 = b1 + 1/10^p;
%t A278811     bn = b1;
%t A278811     For[i = 1, i < Round[n*1.2], ++i,
%t A278811      bn = N[c^(bn/i), acc];
%t A278811      If[bn > th, Break[]];
%t A278811      ];
%t A278811     If[bn > th, {
%t A278811       b1 = b1 - 1/10^p;
%t A278811       Break[];
%t A278811       }];
%t A278811     ];
%t A278811   ];
%t A278811 N[b1,n]
%t A278811 RealDigits[ Fold[ Log[5, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* _Robert G. Wilson v_, Dec 02 2016 *)
%Y A278811 For sequence round(b(n)) see A278451.
%Y A278811 For different values of c see A278808, A278809, A278810, A278812.
%Y A278811 For b(1)=0 see A278813.
%K A278811 nonn,cons
%O A278811 1,3
%A A278811 _Rok Cestnik_, Nov 28 2016