A278811 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278451, where c=5 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

0, 1, 7, 7, 5, 8, 1, 9, 1, 8, 8, 0, 2, 5, 1, 4, 0, 3, 3, 3, 8, 3, 5, 0, 3, 1, 8, 1, 3, 0, 8, 6, 6, 9, 8, 5, 7, 8, 8, 3, 2, 9, 7, 7, 0, 3, 4, 6, 8, 1, 0, 5, 2, 1, 5, 6, 4, 2, 3, 6, 3, 5, 7, 4, 3, 3, 3, 1, 7, 4, 8, 3, 6, 8, 4, 2, 2, 1, 1, 8, 3, 5, 1, 4, 8, 4, 6, 9, 0, 7, 6, 9, 7, 1, 4, 2, 7, 2, 6, 5, 7, 5, 1, 5, 6, 9, 2, 7, 7, 0, 1, 6, 5, 4, 1, 3, 4, 9, 9, 8, 6, 1, 3, 5, 5, 3, 1, 5, 8, 5

Offset

1,3

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=5 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

log5(2*log5(3*log5(4*log5(...)))). - Andrey Zabolotskiy, Nov 30 2016

Example

0.17758191880251403338350318130866985788329770346810...

Mathematica

c = 5;
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[5, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* Robert G. Wilson v, Dec 02 2016 *)

Crossrefs

For sequence round(b(n)) see A278451.

For different values of c see A278808, A278809, A278810, A278812.

For b(1)=0 see A278813.

Sequence in context: A220781 A198566 A154193 * A021932 A244675 A065472

Adjacent sequences: A278808 A278809 A278810 * A278812 A278813 A278814

Keyword

nonn,cons

Author

Rok Cestnik, Nov 28 2016

Status

approved