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A278808
Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278448, where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.
7
2, 8, 7, 1, 8, 8, 0, 8, 2, 7, 0, 4, 5, 4, 5, 4, 6, 5, 8, 8, 9, 0, 5, 5, 1, 7, 5, 5, 0, 4, 5, 7, 5, 0, 4, 5, 8, 6, 5, 6, 5, 2, 5, 1, 1, 8, 4, 7, 9, 6, 5, 6, 5, 6, 7, 9, 2, 9, 9, 5, 4, 0, 1, 0, 8, 4, 0, 4, 5, 7, 9, 6, 8, 3, 0, 8, 9, 2, 7, 0, 3, 6, 0, 1, 8, 2, 8, 6, 3, 8, 1, 8, 6, 7, 6, 7, 8, 7, 5, 4, 8, 0, 8, 4, 3
OFFSET
1,1
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. 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.
No closed form expression is known. Probably transcendental but this is unproved. - Robert G. Wilson v, Dec 01 2016
FORMULA
log_2(2*log_2(3*log_2(4*log_2(...)))). - Andrey Zabolotskiy, Nov 30 2016
EXAMPLE
2.87188082704545465889055175504575045865652511847965...
MATHEMATICA
c = 2;
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[Log2[#1*#2] &, 1, Reverse@Range[2, 144]], 10,
111][[1]] (* Robert G. Wilson v, Dec 01 2016 *)
CROSSREFS
For sequence round(b(n)) see A278448.
For different values of c see A278809, A278810, A278811, A278812.
For b(1)=0 see A278813.
Sequence in context: A336069 A085298 A011060 * A323459 A282089 A195367
KEYWORD
nonn,cons
AUTHOR
Rok Cestnik, Nov 28 2016
STATUS
approved