cp's OEIS Frontend

A278448 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

%I A278448 #25 Dec 02 2016 00:15:08
%S A278448 3,7,13,19,25,32,39,46,53,61,69,77,85,93,102,110,119,128,136,145,154,
%T A278448 163,173,182,191,201,210,220,229,239,248,258,268,278,288,298,307,318,
%U A278448 328,338,348,358,368,379,389,399,410,420,430,441,451,462,473,483,494,505,515,526,537,547,558,569,580,591,602,613,624,635,646,657
%N A278448 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.
%C A278448 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 A278448 In this case b(1) = 2.8718808270... A278808. 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 A278448 The value of b(1) is found through trial and error. 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 A278448 b(n) = n*log_2((n+1)*log_2((n+2)*log_2(...))) ~ n*log_2(n). - _Andrey Zabolotskiy_, Dec 01 2016
%H A278448 Rok Cestnik, <a href="/A278448/b278448.txt">Table of n, a(n) for n = 1..1000</a>
%H A278448 Rok Cestnik, <a href="/A278448/a278448.pdf">Plot of the dependence of b(1) on c</a>
%e A278448 a(2) = round(2^2.87...) = round(7.32...) = 7.
%e A278448 a(3) = round(2^(7.32.../2)) = round(12.64...) = 13.
%e A278448 a(4) = round(2^(12.64.../3)) = round(18.55...) = 19.
%t A278448 c = 2;
%t A278448 n = 100;
%t A278448 acc = Round[n*1.2];
%t A278448 th = 1000000;
%t A278448 b1 = 0;
%t A278448 For[p = 0, p < acc, ++p,
%t A278448   For[d = 0, d < 9, ++d,
%t A278448     b1 = b1 + 1/10^p;
%t A278448     bn = b1;
%t A278448     For[i = 1, i < Round[n*1.2], ++i,
%t A278448      bn = N[c^(bn/i), acc];
%t A278448      If[bn > th, Break[]];
%t A278448      ];
%t A278448     If[bn > th, {
%t A278448       b1 = b1 - 1/10^p;
%t A278448       Break[];
%t A278448       }];
%t A278448     ];
%t A278448   ];
%t A278448 bnlist = {N[b1]};
%t A278448 bn = b1;
%t A278448 For[i = 1, i < n, ++i,
%t A278448   bn = N[c^(bn/i), acc];
%t A278448   If[bn > th, Break[]];
%t A278448   bnlist = Append[bnlist, N[bn]];
%t A278448   ];
%t A278448 anlist = Map[Round[#] &, bnlist]
%Y A278448 For decimal expansion of b(1) see A278808.
%Y A278448 For different values of c see A278449, A278450, A278451, A278452.
%Y A278448 For b(1)=0 see A278453.
%K A278448 nonn
%O A278448 1,1
%A A278448 _Rok Cestnik_, Nov 22 2016