cp's OEIS Frontend

A278808 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278448, where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

%I A278808 #24 Dec 03 2016 12:24:08
%S A278808 2,8,7,1,8,8,0,8,2,7,0,4,5,4,5,4,6,5,8,8,9,0,5,5,1,7,5,5,0,4,5,7,5,0,
%T A278808 4,5,8,6,5,6,5,2,5,1,1,8,4,7,9,6,5,6,5,6,7,9,2,9,9,5,4,0,1,0,8,4,0,4,
%U A278808 5,7,9,6,8,3,0,8,9,2,7,0,3,6,0,1,8,2,8,6,3,8,1,8,6,7,6,7,8,7,5,4,8,0,8,4,3
%N A278808 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278448, where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.
%C A278808 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 A278808 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 A278808 The value of b(1) is found through trial and error. 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.
%C A278808 No closed form expression is known. Probably transcendental but this is unproved. - _Robert G. Wilson v_, Dec 01 2016
%H A278808 Rok Cestnik, <a href="/A278808/b278808.txt">Table of n, a(n) for n = 1..1000</a>
%H A278808 Rok Cestnik, <a href="/A278808/a278808.pdf">Plot of the dependence of b(1) on c</a>
%F A278808 log_2(2*log_2(3*log_2(4*log_2(...)))). - _Andrey Zabolotskiy_, Nov 30 2016
%e A278808 2.87188082704545465889055175504575045865652511847965...
%t A278808 c = 2;
%t A278808 n = 100;
%t A278808 acc = Round[n*1.2];
%t A278808 th = 1000000;
%t A278808 b1 = 0;
%t A278808 For[p = 0, p < acc, ++p,
%t A278808   For[d = 0, d < 9, ++d,
%t A278808     b1 = b1 + 1/10^p;
%t A278808     bn = b1;
%t A278808     For[i = 1, i < Round[n*1.2], ++i,
%t A278808      bn = N[c^(bn/i), acc];
%t A278808      If[bn > th, Break[]];
%t A278808      ];
%t A278808     If[bn > th, {
%t A278808       b1 = b1 - 1/10^p;
%t A278808       Break[];
%t A278808       }];
%t A278808     ];
%t A278808   ];
%t A278808 N[b1,n]
%t A278808 RealDigits[Fold[Log2[#1*#2] &, 1, Reverse@Range[2, 144]], 10,
%t A278808 111][[1]] (* _Robert G. Wilson v_, Dec 01 2016 *)
%Y A278808 For sequence round(b(n)) see A278448.
%Y A278808 For different values of c see A278809, A278810, A278811, A278812.
%Y A278808 For b(1)=0 see A278813.
%K A278808 nonn,cons
%O A278808 1,1
%A A278808 _Rok Cestnik_, Nov 28 2016