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.

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, 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, 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

Offset

1,1

Comments

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.

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.

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.

No closed form expression is known. Probably transcendental but this is unproved. - Robert G. Wilson v, Dec 01 2016

Links

Rok Cestnik, Table of n, a(n) for n = 1..1000

Rok Cestnik, Plot of the dependence of b(1) on c

Formula

log_2(2*log_2(3*log_2(4*log_2(...)))). - Andrey Zabolotskiy, Nov 30 2016

Example

2.87188082704545465889055175504575045865652511847965...

Mathematica

c = 2;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
N[b1, n]
RealDigits[Fold[Log2[#1*#2] &, 1, Reverse@Range[2, 144]], 10,
111][[1]] (* Robert G. Wilson v, Dec 01 2016 *)

Crossrefs

For sequence round(b(n)) see A278448.

For different values of c see A278809, A278810, A278811, A278812.

For b(1)=0 see A278813.

Sequence in context: A336069 A085298 A011060 * A323459 A282089 A195367

Adjacent sequences: A278805 A278806 A278807 * A278809 A278810 A278811

Keyword

nonn,cons

Author

Rok Cestnik, Nov 28 2016

Status

approved