A225921 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+6) and a(1) = 1.

1, 14, 50, 150, 427, 1195, 3324, 9226, 25587, 70942, 196672, 545212, 1511411, 4189842, 11614806, 32197786, 89256522, 247430866, 685911016, 1901435842, 5271031028, 14611993445, 40506373648, 112289011899, 311279955644, 862909105217, 2392097886032, 6631210937220, 18382591594862

Offset

1,2

Comments

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

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

Links

Table of n, a(n) for n=1..29.

Formula

For n>=3, a(n) = ceiling( (a(n-1)+6.5)^2 / (a(n-2)+6.5) - 6.5 ) unless the fractional part of the number inside ceiling() is very small (~ 1/a(n-2)). - Max Alekseyev, Jan 27 2022

Example

a(1) = 1 by decree; a(2) = 14 because 1/8 + ... + 1/19 < 1 < 1/8 + ... + 1/(14+6), so that a(3) = 50 because 1/21 + ... + 1/55 < 1/8 + ... + 1/20 < 1/21 + ... + 1/(50+6).
Successive values of a(n) yield a chain: 1 < 1/(1+7) + ... + 1/(14+6) < 1/(14+7) + ... + 1/(50+6) < 1/(50+7) + ... + 1/(150+6) < ...
Abbreviating this chain as b(1) = 1 < b(2) < b(3) < b(4) < ... < R = 2.7721..., it appears that lim_{n->infinity} b(n) = log(R) = 1.0196... .

Mathematica

nn = 11; f[n_] := 1/(n + 6); 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]]

Crossrefs

Cf. A225918.

Sequence in context: A009960 A009928 A050441 * A350850 A205354 A281699

Adjacent sequences: A225918 A225919 A225920 * A225922 A225923 A225924

Keyword

nonn

Author

Clark Kimberling, May 21 2013

Extensions

a(13)-a(16) from Robert G. Wilson v, May 22 2013

a(17)-a(18) from Robert G. Wilson v, Jun 13 2013

a(19) from Jinyuan Wang, Jun 14 2020

Terms a(20) on from Max Alekseyev, Jan 27 2022

Status

approved