cp's OEIS Frontend

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.

%I A278450 #21 Dec 02 2016 00:15:54
%S A278450 0,2,4,6,9,12,14,17,21,24,27,31,34,38,41,45,49,52,56,60,64,68,72,76,
%T A278450 80,84,88,92,96,101,105,109,114,118,122,127,131,135,140,144,149,153,
%U A278450 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
%N 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.
%C A278450 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).
%C A278450 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.
%C A278450 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."
%C A278450 b(n) = n*log_4((n+1)*log_4((n+2)*log_4(...))) ~ n*log_4(n). - _Andrey Zabolotskiy_, Dec 01 2016
%H A278450 Rok Cestnik, <a href="/A278450/b278450.txt">Table of n, a(n) for n = 1..1000</a>
%H A278450 Rok Cestnik, <a href="/A278450/a278450.pdf">Plot of the dependence of b(1) on c</a>
%e A278450 a(2) = round(4^0.49...) = round(1.99...) = 2.
%e A278450 a(3) = round(4^(1.99.../2)) = round(3.97...) = 4.
%e A278450 a(4) = round(4^(3.97.../3)) = round(6.28...) = 6.
%t A278450 c = 4;
%t A278450 n = 100;
%t A278450 acc = Round[n*1.2];
%t A278450 th = 1000000;
%t A278450 b1 = 0;
%t A278450 For[p = 0, p < acc, ++p,
%t A278450   For[d = 0, d < 9, ++d,
%t A278450     b1 = b1 + 1/10^p;
%t A278450     bn = b1;
%t A278450     For[i = 1, i < Round[n*1.2], ++i,
%t A278450      bn = N[c^(bn/i), acc];
%t A278450      If[bn > th, Break[]];
%t A278450      ];
%t A278450     If[bn > th, {
%t A278450       b1 = b1 - 1/10^p;
%t A278450       Break[];
%t A278450       }];
%t A278450     ];
%t A278450   ];
%t A278450 bnlist = {N[b1]};
%t A278450 bn = b1;
%t A278450 For[i = 1, i < n, ++i,
%t A278450   bn = N[c^(bn/i), acc];
%t A278450   If[bn > th, Break[]];
%t A278450   bnlist = Append[bnlist, N[bn]];
%t A278450   ];
%t A278450 anlist = Map[Round[#] &, bnlist]
%Y A278450 For decimal expansion of b(1) see A278810.
%Y A278450 For different values of c see A278448, A278449, A278451, A278452.
%Y A278450 For b(1)=0 see A278453.
%K A278450 nonn
%O A278450 1,2
%A A278450 _Rok Cestnik_, Nov 22 2016