A248633 Least k such that 20/27- sum{(h^2)/4^h, h = 1..k} < 1/8^n.

3, 5, 7, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 36, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104

Offset: 1

Author

Clark Kimberling, Oct 11 2014

Keywords

• nonn
• easy

Comments

This sequence provides insight into the manner of convergence of sum{(h^2)/4^h, h = 1..k} to 20/27. The difference sequence of A248633 entirely of 1s and 2s, so that A248634 and A248635 partition the positive integers.

Examples

			Let s(n) = 20/27 - sum{(h^2)/4^h, h = 1..n}.  Approximations follow:
n ... s(n) ........ 1/8^n
1 ... 0.49074 ... 0.125000
2 ... 0.24074 ... 0.015625
3 ... 0.10011 ... 0.001953
4 ... 0.03761 ... 0.000244
5 ... 0.01320 ... 0.000030
a(2) = 5 because s(5) < 1/8^2 < s(2).
		

Links

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

Crossrefs

Cf. A248632, A248630.

Programs

Mathematica

z= 300; p[k_] := p[k] = Sum[(h^2/4^h), {h, 1, k}];
d = N[Table[20/27 - p[k], {k, 1, z/5}], 12];
f[n_] := f[n] = Select[Range[z], 20/27 - p[#] < 1/8^n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]  (* A248633 *)
d = Differences[u]
Flatten[Position[d, 1]]  (* A248634 *)
Flatten[Position[d, 2]]  (* A248635 *)