A278451 - OEIS

%I #21 Dec 02 2016 00:16:20

%S 0,1,3,5,7,9,11,14,17,19,22,25,28,31,34,37,40,43,46,49,52,56,59,62,66,

%T 69,73,76,80,83,87,90,94,98,101,105,109,112,116,120,123,127,131,135,

%U 139,143,146,150,154,158,162,166,170,174,178,182,186,190,194,198,202,206,210,214,218,222,226,231,235,239

%N a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=5 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

%C 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 In this case b(1) = 0.1775819188... A278811. 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 The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=5 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 b(n) = n*log_5((n+1)*log_5((n+2)*log_5(...))) ~ n*log_5(n). - _Andrey Zabolotskiy_, Dec 01 2016

%H Rok Cestnik, <a href="/A278451/b278451.txt">Table of n, a(n) for n = 1..1000</a>

%H Rok Cestnik, <a href="/A278451/a278451.pdf">Plot of the dependence of b(1) on c</a>

%e a(2) = round(5^0.17...) = round(1.33...) = 1.

%e a(3) = round(5^(1.33.../2)) = round(2.91...) = 3.

%e a(4) = round(5^(2.91.../3)) = round(4.78...) = 5.

%t c = 5;

%t n = 100;

%t acc = Round[n*1.2];

%t th = 1000000;

%t b1 = 0;

%t For[p = 0, p < acc, ++p,

%t   For[d = 0, d < 9, ++d,

%t     b1 = b1 + 1/10^p;

%t     bn = b1;

%t     For[i = 1, i < Round[n*1.2], ++i,

%t      bn = N[c^(bn/i), acc];

%t      If[bn > th, Break[]];

%t      ];

%t     If[bn > th, {

%t       b1 = b1 - 1/10^p;

%t       Break[];

%t       }];

%t     ];

%t   ];

%t bnlist = {N[b1]};

%t bn = b1;

%t For[i = 1, i < n, ++i,

%t   bn = N[c^(bn/i), acc];

%t   If[bn > th, Break[]];

%t   bnlist = Append[bnlist, N[bn]];

%t   ];

%t anlist = Map[Round[#] &, bnlist]

%Y For decimal expansion of b(1) see A278811.

%Y For different values of c see A278448, A278449, A278450, A278452.

%Y For b(1)=0 see A278453.

%K nonn

%O 1,3

%A _Rok Cestnik_, Nov 22 2016