

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
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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(n1)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(n1)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



