cp's OEIS Frontend

A278812 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278452, where c = e = 2.71828... and b(1) is chosen such that the sequence neither explodes nor goes to 1.

%I A278812 #19 Jul 04 2017 16:23:13
%S A278812 1,3,6,7,9,0,1,2,6,1,7,9,7,0,8,5,1,6,9,6,6,8,9,0,9,1,7,5,7,6,0,4,8,8,
%T A278812 5,3,8,3,8,4,6,2,4,5,2,6,1,8,2,1,3,5,7,7,0,4,1,4,6,0,3,7,1,3,8,6,3,3,
%U A278812 1,7,9,4,4,8,8,0,1,5,6,8,6,5,6,6,7,1,5,8,8,6,8,3,7,2,7,7,3,7,4,9,5,6,2,4,7,7,4,3,3,4,9,8,1,9,3,3,3,6,1,7,1,9,6,1,1,1,3,2,2,8
%N A278812 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278452, where c = e = 2.71828... and b(1) is chosen such that the sequence neither explodes nor goes to 1.
%C A278812 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 A278812 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 A278812 The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=e 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 A278812 Rok Cestnik, <a href="/A278812/b278812.txt">Table of n, a(n) for n = 1..1000</a>
%H A278812 Rok Cestnik, <a href="/A278812/a278812.pdf">Plot of the dependence of b(1) on c</a>
%F A278812 log(2*log(3*log(4*log(...)))). - _Andrey Zabolotskiy_, Nov 30 2016
%e A278812 1.36790126179708516966890917576048853838462452618213...
%t A278812 c = E;
%t A278812 n = 100;
%t A278812 acc = Round[n*1.2];
%t A278812 th = 1000000;
%t A278812 b1 = 0;
%t A278812 For[p = 0, p < acc, ++p,
%t A278812   For[d = 0, d < 9, ++d,
%t A278812     b1 = b1 + 1/10^p;
%t A278812     bn = b1;
%t A278812     For[i = 1, i < Round[n*1.2], ++i,
%t A278812      bn = N[c^(bn/i), acc];
%t A278812      If[bn > th, Break[]];
%t A278812      ];
%t A278812     If[bn > th, {
%t A278812       b1 = b1 - 1/10^p;
%t A278812       Break[];
%t A278812       }];
%t A278812     ];
%t A278812   ];
%t A278812 N[b1,n]
%t A278812 RealDigits[ Fold[ Log[#1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* _Robert G. Wilson v_, Dec 02 2016 *)
%Y A278812 For sequence round(b(n)) see A278452.
%Y A278812 For different values of c see A278808, A278809, A278810, A278811.
%Y A278812 For b(1)=0 see A278813.
%K A278812 nonn,cons
%O A278812 1,2
%A A278812 _Rok Cestnik_, Nov 28 2016