

A278453


a(n) = nearest integer to b(n) = c^(b(n1)/(n1)), where b(1)=0 and c is chosen such that the sequence neither explodes nor goes to 1.


7



0, 1, 2, 4, 6, 8, 10, 12, 15, 17, 19, 22, 25, 27, 30, 33, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 75, 78, 81, 84, 88, 91, 94, 98, 101, 104, 108, 111, 114, 118, 121, 125, 128, 132, 135, 139, 142, 146, 149, 153, 157, 160, 164, 167, 171, 175, 178, 182, 186, 189, 193, 197, 201, 204, 208, 212, 216
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OFFSET

1,3


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(n1)b(n+1)/b(n)^2 < 1 (due to rounding to the nearest integer a(n1)a(n+1)/a(n)^2 is not always less than 1).
Its value: c = 5.7581959391... A278813. If c were chosen smaller the sequence would approach 1, if it were chosen greater the sequence would at some point violate b(n1)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 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.
b(n) = n*log_c((n+1)*log_c((n+2)*log_c(...))). At n=1 this gives the relation between c and b(1). It follows that b(n) ~ n*log_c(n).  Andrey Zabolotskiy, Nov 30 2016


LINKS

Rok Cestnik, Table of n, a(n) for n = 1..1000
Rok Cestnik, Plot of the dependence of b(1) on c


FORMULA

a(n) = round(n*log_c((n+1)*log_c((n+2)*log_c(...)))).  Andrey Zabolotskiy, Nov 30 2016


EXAMPLE

a(2) = round(5.75...^0) = round(1) = 1.
a(3) = round(5.75...^(1/2)) = round(2.39...) = 2.
a(4) = round(5.75...^(2.39.../3)) = round(4.05...) = 4.


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[];
}];
];
];
bnlist = {N[b1]};
bn = b1;
For[i = 1, i < n, ++i,
bn = N[c^(bn/i), acc];
If[bn > th, Break[]];
bnlist = Append[bnlist, N[bn]];
];
anlist = Map[Round[#] &, bnlist]


CROSSREFS

For decimal expansion of c see A278813.
For different values of b(1) see A278448, A278449, A278450, A278451, A278452.
Sequence in context: A292651 A113903 A248563 * A347009 A130261 A186384
Adjacent sequences: A278450 A278451 A278452 * A278454 A278455 A278456


KEYWORD

nonn


AUTHOR

Rok Cestnik, Nov 22 2016


STATUS

approved



