login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A225920 a(n) is the least k such that f(a(n-1)+1) + ... + f(k) > f(a(n-2)+1) + ... + f(a(n-1)) for n > 1, where f(n) = 1/(n+5) and a(1) = 1. 1

%I #23 Jan 27 2022 20:38:09

%S 1,13,48,150,447,1312,3831,11167,32531,94748,275938,803605,2340292,

%T 6815476,19848236,57802615,168334451,490228448,1427657419,4157665074,

%U 12108072013,35261476137,102689486632,299055281267,870917405325,2536310757258,7386317253546,21510645891422

%N a(n) is the least k such that f(a(n-1)+1) + ... + f(k) > f(a(n-2)+1) + ... + f(a(n-1)) for n > 1, where f(n) = 1/(n+5) and a(1) = 1.

%C Conjecture: a(n) is linearly recurrent. See A225918 for details.

%C The sequence does not satisfy any linear recurrence of order below 50, which suggests it's unlikely to exist. - _Max Alekseyev_, Jan 27 2022

%F For n>=3, a(n) = ceiling( (a(n-1)+5.5)^2 / (a(n-2)+5.5) - 5.5 ) unless the fractional part of the number inside ceiling() is very small (~ 1/a(n-2)). - _Max Alekseyev_, Jan 27 2022

%e a(1) = 1 by decree; a(2) = 13 because 1/7 + ... + 1/17 < 1 < 1/7 + ... + 1/(13+5), so that a(3) = 48 because 1/19 + ... + 1/52 < 1/7 + ... + 1/18 < 1/19 + ... + 1/(48+5).

%e Successive values of a(n) yield a chain: 1 < 1/(1+6) + ... + 1/(13+5) < 1/(13+6) + ... + 1/(48+5) < 1/(48+6) + ... + 1/(150+5) < ...

%e Abbreviating this chain as b(1) = 1 < b(2) < b(3) < b(4) < ... < R = 2.912229..., it appears that lim_{n->infinity} b(n) = log(R) = 1.068918... .

%t nn = 11; f[n_] := 1/(n + 5); a[1] = 1; g[n_] := g[n] = Sum[f[k], {k, 1, n}]; s = 0; a[2] = NestWhile[# + 1 &, 2, ! (s += f[#]) >= a[1] &]; s = 0; a[3] = NestWhile[# + 1 &, a[2] + 1, ! (s += f[#]) >= g[a[2]] - f[1] &]; Do[s = 0; a[z] = NestWhile[# + 1 &, a[z - 1] + 1, ! (s += f[#]) >= g[a[z - 1]] - g[a[z - 2]] &], {z, 4, nn}]; m = Map[a, Range[nn]]

%Y Cf. A225918.

%K nonn

%O 1,2

%A _Clark Kimberling_, May 21 2013

%E a(12)-a(16) from _Robert G. Wilson v_, May 22 2013

%E a(17)-a(18) from _Robert G. Wilson v_, Jun 13 2013

%E a(18) corrected by and a(19) from _Jinyuan Wang_, Jun 14 2020

%E Terms a(20) on from _Max Alekseyev_, Jan 27 2022

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)