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A278810 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. 7
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
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=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."

LINKS

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

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

FORMULA

log4(2*log4(3*log4(4*log4(...)))). - Andrey Zabolotskiy, Nov 30 2016

EXAMPLE

0.49704500007589450773783761552966893614239324798593...

MATHEMATICA

c = 4;

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[4, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* Robert G. Wilson v, Dec 02 2016 *)

CROSSREFS

For sequence round(b(n)) see A278450.

For different values of c see A278808, A278809, A278811, A278812.

For b(1)=0 see A278813.

Sequence in context: A130041 A109987 A021672 * A176426 A114720 A053511

Adjacent sequences:  A278807 A278808 A278809 * A278811 A278812 A278813

KEYWORD

nonn,cons

AUTHOR

Rok Cestnik, Nov 28 2016

STATUS

approved

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Last modified July 31 17:15 EDT 2021. Contains 346376 sequences. (Running on oeis4.)