|
%I #15 Dec 03 2016 12:24:00
%S 0,4,9,7,0,4,5,0,0,0,0,7,5,8,9,4,5,0,7,7,3,7,8,3,7,6,1,5,5,2,9,6,6,8,
%T 9,3,6,1,4,2,3,9,3,2,4,7,9,8,5,9,3,8,9,5,9,8,3,0,3,6,8,4,6,1,2,7,6,0,
%U 5,6,6,4,4,3,1,8,7,3,5,5,7,9,7,8,8,3,6,3,2,4,9,8,4,6,7,7,2,1,6,2,5,2,9,5,7,5,7,6,5,3,0,8,0,1,4,5,3,8,6,4,1,6,3,9,7,6,9,8,9,3
%N Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278450, where c=4 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=4 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="/A278810/b278810.txt">Table of n, a(n) for n = 1..1000</a>
%H Rok Cestnik, <a href="/A278810/a278810.pdf">Plot of the dependence of b(1) on c</a>
%F log4(2*log4(3*log4(4*log4(...)))). - _Andrey Zabolotskiy_, Nov 30 2016
%e 0.49704500007589450773783761552966893614239324798593...
%t c = 4;
%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[4, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* _Robert G. Wilson v_, Dec 02 2016 *)
%Y For sequence round(b(n)) see A278450.
%Y For different values of c see A278808, A278809, A278811, A278812.
%Y For b(1)=0 see A278813.
%K nonn,cons
%O 1,2
%A _Rok Cestnik_, Nov 28 2016
|
