A248631 Least k such that 3/2 - sum{(h^2)/3^h, h = 1..k} < 1/2^n.

3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47, 47, 48, 49, 49

Offset

1,1

Comments

This sequence provides insight into the manner of convergence of sum{(h^2)/3^h, h = 1..k} to 3/2.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

Let s(n) = 3/2 - sum{(h^2)/3^h, h = 1..n}.  Approximations follow:
n ... s(n) ...... 1/2^n
1 ... 1.16666 ... 0.500000
2 ... 0.72222 ... 0.250000
3 ... 0.38888 ... 0.125000
4 ... 0.03909 ... 0.062500
5 ... 0.08847 ... 0.031250
6 ... 0.03909 ... 0.015625
7 ... 0.01668 ... 0.007812
a(5) = 7 because s(7) < 1/32 < s(6).

Mathematica

z = 200; p[k_] := p[k] = Sum[(h^2/2^h), {h, 1, k}];
d = N[Table[6 - p[k], {k, 1, z/5}], 12];
f[n_] := f[n] = Select[Range[z], 6 - p[#] < 1/3^n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]; (* A248629 *)
d = Differences[u];
Flatten[Position[d, 1]];  (* A248630 *)

Crossrefs

Cf. A248632, A248630.

Sequence in context: A202307 A009041 A196033 * A176407 A077396 A243196

Adjacent sequences: A248628 A248629 A248630 * A248632 A248633 A248634

Keyword

nonn,easy

Author

Clark Kimberling, Oct 11 2014

Status

approved