OFFSET
0,4
COMMENTS
This sequence illustrates the second law of small numbers because it is a coincidence that the terms for 1 <= n <= 8 are the same as the Fibonacci numbers F(n) (A000045): a(n) = F(n) for 1 <= n <= 8.
Furthermore, the following terms are the sum of two Fibonacci numbers: a(9) = F(9) + F(2), a(10) = F(10) + F(4), a(11) = F(11) + F(6), a(14) = F(14) + F(11); or the algebraic sum of three Fibonacci numbers: a(12) = F(12) + F(8) - F(3), a(13) = F(13) + F(10) - F(7), a(14) = F(14) + F(12) - F(10), a(18) = F(19) - F(13) - F(8), a(19) = F(20) + F(10) - F(4); or the algebraic sum of four Fibonacci numbers: a(15) = F(15) + F(12) + F(9) + F(5), a(16) = F(16) + F(14) - F(6) - F(4), a(17) = F(18) - F(13) - F(9) - F(3), a(18) = F(18) + F(16) + F(14) + F(8), a(19) = F(19) + F(18) + F(10) - F(3).
Note that, for following values of n, a(n) > F(n+1) for n >= 20.
Remark as well that (2^(1/2) + 3^(1/2)) = 3.14626437... ~= Pi (see A135611).
REFERENCES
T. Koshy, Fibonacci and Lucas Numbers with Applications, New York, Wiley-Interscience, 2001
I. Stewart, L'univers des nombres, 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.
I. Stewart, Fibonacci Forgeries
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Strong Law of Small Numbers.
FORMULA
a(n) = round(2^((n-3)/2) + 3^((n-3)/2)).
MAPLE
seq(round(2^((n-3)/2)+3^((n-3)/2)), n=0..50);
MATHEMATICA
Table[Round[2^((n - 3)/2) + 3^((n - 3)/2)], {n, 0, 50}] (* Vincenzo Librandi, Sep 20 2013 *)
PROG
(Magma) [Round(2^((n-3)/2) + 3^((n-3)/2)): n in [0..50]]; // Vincenzo Librandi, Sep 20 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Pletser, Sep 15 2013
STATUS
approved