A247911 - OEIS

A247911

Least number k such that (2*k+1)/u(2*k+1) - e < 1/n^n, where u is defined as in Comments.

1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 39, 39, 40, 41, 41

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OFFSET

1,2

COMMENTS

The sequence u is define recursively by u(n) = u(n-1) + u(n-2)/(n-2), with u(1) = 0 and u(2) = 1. Let d(n) = a(n+1) - a(n). It appears that d(n) is in {2,3} for n >= 1, that d(n+1) - d(n) is in {-1,0,1}, and that similar bounds hold for higher differences.

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 19.

LINKS

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

EXAMPLE

Approximations for the first few terms (2*n+1)/u(2*n+1) - e and 1/n^n are shown here:

n ... (2*n+1)/u(2*n+1)-e .. 1/n^n

1 ... 0.28171817 .......... 1

2 ... 0.0089908988 ........ 0.25

3 ... 0.0001647734 ........ 0.037037

4 ... 0.0000018654 ........ 0.00390625

5 ... 0.0000000143 ........ 0.00032000