A225922 - OEIS

%I #21 Jan 27 2022 20:45:49

%S 1,16,58,176,507,1436,4043,11359,31890,89506,251193,704933,1978258,

%T 5551574,15579326,43720081,122691130,344306598,966223316,2711500429,

%U 7609249779,21353742568,59924740904,168166051476,471922288540,1324349620283,3716505787761,10429583743512

%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+7) 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)+7.5)^2 / (a(n-2)+7.5) - 7.5 ) 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) = 15 because 1/9 + ... + 1/21 < 1 < 1/9 + ... + 1/(15+7), so that a(3) = 58 because 1/23 + ... + 1/57 < 1/9 + ... + 1/22 < 1/23 + ... + 1/(58+7).

%e Successive values of a(n) yield a chain: 1 < 1/(1+8) + ... + 1/(15+7) < 1/(15+8) + ... + 1/(58+7) < 1/(58+8) + ... + 1/(176+7) < ...

%e Abbreviating this chain as b(1) = 1 < b(2) < b(3) < b(4) < ... < R = 2.80628..., it appears that lim_{n->infinity} b(n) = log(R) = 1.03186... .

%t nn = 11; f[n_] := 1/(n + 7); 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(10)-a(17) from _Robert G. Wilson v_, May 22 2013

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

%E a(19) from _Jinyuan Wang_, Jun 14 2020

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