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A278450 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=4 and b(1) is chosen such that the sequence neither explodes nor goes to 1. 7
0, 2, 4, 6, 9, 12, 14, 17, 21, 24, 27, 31, 34, 38, 41, 45, 49, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 101, 105, 109, 114, 118, 122, 127, 131, 135, 140, 144, 149, 153, 158, 162, 167, 172, 176, 181, 185, 190, 195, 200, 204, 209, 214, 218, 223, 228, 233, 238, 242, 247, 252, 257, 262, 267, 272, 277, 282, 287 (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) = 0.4970450000... A278810. 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=4 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_4((n+1)*log_4((n+2)*log_4(...))) ~ n*log_4(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(4^0.49...) = round(1.99...) = 2.

a(3) = round(4^(1.99.../2)) = round(3.97...) = 4.

a(4) = round(4^(3.97.../3)) = round(6.28...) = 6.

MATHEMATICA

c = 4;

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 A278810.

For different values of c see A278448, A278449, A278451, A278452.

For b(1)=0 see A278453.

Sequence in context: A183422 A025057 A189753 * A030763 A143145 A256393

Adjacent sequences:  A278447 A278448 A278449 * A278451 A278452 A278453

KEYWORD

nonn

AUTHOR

Rok Cestnik, Nov 22 2016

STATUS

approved

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Last modified July 16 12:19 EDT 2019. Contains 325079 sequences. (Running on oeis4.)