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A278449 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=3 and b(1) is chosen such that the sequence neither explodes nor goes to 1. 7
1, 3, 6, 9, 13, 16, 20, 24, 29, 33, 37, 42, 47, 51, 56, 61, 66, 71, 76, 81, 86, 92, 97, 102, 108, 113, 118, 124, 129, 135, 141, 146, 152, 158, 163, 169, 175, 181, 187, 193, 199, 205, 210, 216, 222, 229, 235, 241, 247, 253, 259, 265, 271, 278, 284, 290, 296, 303, 309, 315, 322, 328, 334, 341, 347, 354, 360, 367, 373, 379 (list; graph; refs; listen; history; text; internal format)
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(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) = 1.0828736095... A278809. 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=3 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_3((n+1)*log_3((n+2)*log_3(...))) ~ n*log_3(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(3^1.08...) = round(3.28...) = 3.

a(3) = round(3^(3.28.../2)) = round(6.07...) = 6.

a(4) = round(3^(6.07.../3)) = round(9.26...) = 9.

MATHEMATICA

c = 3;

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 A278809.

For different values of c see A278448, A278450, A278451, A278452.

For b(1)=0 see A278453.

Sequence in context: A289037 A060605 A325228 * A006590 A061781 A123753

Adjacent sequences: A278446 A278447 A278448 * A278450 A278451 A278452

KEYWORD

nonn

AUTHOR

Rok Cestnik, Nov 22 2016

STATUS

approved

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Last modified January 27 10:15 EST 2023. Contains 359838 sequences. (Running on oeis4.)