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A278813 Decimal expansion of c in the sequence b(n+1) = c^(b(n)/n) A278453, where b(1)=0 and c is chosen such that the sequence neither explodes nor goes to 1. 7
5, 7, 5, 8, 1, 9, 5, 9, 3, 9, 1, 1, 0, 3, 7, 4, 9, 4, 1, 9, 7, 4, 0, 2, 8, 8, 6, 5, 0, 0, 9, 3, 2, 9, 0, 9, 2, 4, 7, 4, 2, 4, 2, 6, 4, 7, 0, 5, 5, 3, 1, 5, 4, 1, 5, 1, 4, 1, 2, 5, 9, 9, 0, 6, 1, 9, 7, 1, 0, 7, 5, 9, 8, 9, 1, 5, 8, 7, 2, 3, 0, 8, 3, 3, 3, 7, 8, 7, 0, 6, 9, 5, 8, 7, 9, 1, 1, 5, 7, 2, 0, 0, 5, 6, 2, 9, 5, 0, 5, 6, 3, 2, 1, 1, 0, 5, 7, 1, 4, 7, 1, 3, 5, 9, 5, 0, 6, 0, 7, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

There exists a unique value of c 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 c 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 c is found through trial and error. Suppose one starts with c = 5, the sequence b(n) would continue b(2) = 1, b(3) = 2.23..., b(4) = 3.31..., b(5) = 3.80..., b(6) = 3.39..., b(7) = 2.48..., b(8) = 1.77... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say c = 6, which gives rise to b(2) = 1, b(3) = 2.44, b(4) = 4.31..., b(5) = 6.92..., b(6) = 11.94..., b(7) = 35.38... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of c is between 5 and 6.

c satisfies 2*log_c(3*log_c(4*log_c(...))) = 1. - Andrey Zabolotskiy, Dec 02 2016

No closed form expression is known. Probably transcendental but this is unproved. - Robert G. Wilson v, Dec 02 2016

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..2500 First 1000 from Rok Cestnik.

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

EXAMPLE

5.75819593911037494197402886500932909247424264705531...

MATHEMATICA

b1 = 0;

n = 100;

acc = Round[n*1.2];

th = 1000000;

c = 0;

For[p = 0, p < acc, ++p, For[d = 0, d < 9, ++d, c = c + 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, {c = c - 1/10^p;

      Break[];

      }];

    ];

  ];

N[c, n]

CROSSREFS

For sequence round(b(n)) see A278453.

For different values of b(1) see A278808, A278809, A278810, A278811, A278812.

Sequence in context: A010718 A247872 A090987 * A217167 A195498 A065746

Adjacent sequences:  A278810 A278811 A278812 * A278814 A278815 A278816

KEYWORD

nonn,cons,nice

AUTHOR

Rok Cestnik, Nov 28 2016

STATUS

approved

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Last modified October 23 22:04 EDT 2018. Contains 316541 sequences. (Running on oeis4.)