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.

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, 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, 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

Offset

1,2

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. 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."

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

Example

1.36790126179708516966890917576048853838462452618213...

Mathematica

c = E;
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[ Log[#1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* Robert G. Wilson v, Dec 02 2016 *)

Crossrefs

For sequence round(b(n)) see A278452.

For different values of c see A278808, A278809, A278810, A278811.

For b(1)=0 see A278813.

Sequence in context: A310133 A346878 A034091 * A111717 A181911 A181919

Adjacent sequences: A278809 A278810 A278811 * A278813 A278814 A278815

Keyword

nonn,cons

Author

Rok Cestnik, Nov 28 2016

Status

approved