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A278451 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=5 and b(1) is chosen such that the sequence neither explodes nor goes to 1. 7
0, 1, 3, 5, 7, 9, 11, 14, 17, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 56, 59, 62, 66, 69, 73, 76, 80, 83, 87, 90, 94, 98, 101, 105, 109, 112, 116, 120, 123, 127, 131, 135, 139, 143, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 231, 235, 239 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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 (due to rounding to the nearest integer a(n-1)a(n+1)/a(n)^2 is not always less than 1).

In this case b(1) = 0.1775819188... A278811. If b(1) were chosen smaller the sequence would approach 1, if it were chosen greater the sequence 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=5 similar): "Suppose one starts with b(1) = 2, the sequence 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."

b(n) = n*log_5((n+1)*log_5((n+2)*log_5(...))) ~ n*log_5(n). - Andrey Zabolotskiy, Dec 01 2016

LINKS

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

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

EXAMPLE

a(2) = round(5^0.17...) = round(1.33...) = 1.

a(3) = round(5^(1.33.../2)) = round(2.91...) = 3.

a(4) = round(5^(2.91.../3)) = round(4.78...) = 5.

MATHEMATICA

c = 5;

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[];

}];

];

];

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 b(1) see A278811.

For different values of c see A278448, A278449, A278450, A278452.

For b(1)=0 see A278453.

Sequence in context: A291839 A134917 A066665 * A175269 A318919 A279539

Adjacent sequences: A278448 A278449 A278450 * A278452 A278453 A278454

KEYWORD

nonn

AUTHOR

Rok Cestnik, Nov 22 2016

STATUS

approved

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Last modified November 28 09:59 EST 2022. Contains 358411 sequences. (Running on oeis4.)