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A278453 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), 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 (list; graph; refs; listen; history; text; internal format)
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(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).

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(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 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

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Last modified February 7 15:57 EST 2023. Contains 360127 sequences. (Running on oeis4.)