A005181 a(n) = ceiling(exp((n-1)/2)).

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 91, 149, 245, 404, 666, 1097, 1809, 2981, 4915, 8104, 13360, 22027, 36316, 59875, 98716, 162755, 268338, 442414, 729417, 1202605, 1982760, 3269018, 5389699, 8886111, 14650720, 24154953, 39824785, 65659970, 108254988, 178482301

Offset: 0

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn

Comments

This sequence illustrates the second law of small numbers because it is a coincidence that its first ten terms are the same as the first ten Fibonacci numbers (A000045). - Alonso del Arte, Mar 18 2013

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• I. Stewart, L'univers des nombres, pp. 27 Belin-Pour La Science, Paris 2000.

Links

• Vladimir Pletser, Table of n, a(n) for n = 0..1000
• R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
• R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
• R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
• I. Stewart, Fibonacci Forgeries
• Eric Weisstein's World of Mathematics, Fibonacci Number.
• Eric Weisstein's World of Mathematics, Strong Law of Small Numbers.

Crossrefs

Cf. A000045, A019774.

Programs

Maple

seq(round(ceil(exp((n-1)/2))), n=0..50); # Vladimir Pletser, Sep 15 2013

Mathematica

Table[Ceiling[E^((n - 1)/2)], {n, 0, 39}] (* Alonso del Arte, Mar 18 2013 *)

Python

import math
for n in range(99):
    print(str(int(math.ceil(math.e**((n-1)*0.5)))), end=', ')
# Alex Ratushnyak, Mar 18 2013

Formula

Limit_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774. - Alois P. Heinz, Feb 19 2019

Extensions

A few more terms from Alonso del Arte, Mar 18 2013