A248103 Least k such that ((2k+1)/(2k-1))^k < 1/(2n^2).

1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 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, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 46

Offset

1,2

Comments

In general, for fixed positive m, the limit of ((m*x+1)/(m*x-1))^x is e^(2/m), as illustrated by A248103, A248106, A248111.

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 14.

Links

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

Example

Approximations are shown here:
n ... ((2n+1)/(2n-1))^n ... 1/(2*n^2)
1 ... 0.281718 ............ 0.5
2 ... 0.0595959 ........... 0.125
3 ... 0.0257182 ........... 0.05555
4 ... 0.0143296 ........... 0.3125
a(4) = 3 because p(4) - e < 1/32 < p(3) - e.

Mathematica

z = 1200; p[k_] := p[k] = ((2 k + 1)/(2 k - 1))^k;
N[Table[p[n] - E, {n, 1, z/8}]]
f[n_] := f[n] = Select[Range[z], p[#] - E < 1/(2 n^2) &, 1]
u = Flatten[Table[f[n], {n, 1, z/10}]]  (* A248103 *)

Crossrefs

Cf. A248106, A248111.

Sequence in context: A131737 A004396 A066481 * A121928 A175406 A210435

Adjacent sequences: A248100 A248101 A248102 * A248104 A248105 A248106

Keyword

nonn,easy

Author

Clark Kimberling, Oct 02 2014

Status

approved