A229194 - OEIS

%I #23 Sep 08 2022 08:46:05

%S 1,1,1,2,3,5,8,13,21,35,58,97,163,275,466,793,1353,2315,3969,6817,

%T 11726,20195,34816,60073,103724,179195,309724,535537,926275,1602515,

%U 2773034,4799353,8307516,14381675,24899377,43112257,74651790,129271235,223862687,387682633,671402698,1162785755,2013837368,3487832977,6040770648,10462450355,18120829034,31385253913,54359521280,94151567435,163072632198

%N Integers nearest to (2^((n-3)/2) + 3^((n-3)/2)).

%C 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.

%C 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).

%C Note that, for following values of n, a(n) > F(n+1) for n >= 20.

%C Remark as well that (2^(1/2) + 3^(1/2)) = 3.14626437... ~=  Pi (see A135611).

%D T. Koshy, Fibonacci and Lucas Numbers with Applications, New York, Wiley-Interscience, 2001

%D I. Stewart, L'univers des nombres, Belin-Pour La Science, Paris 2000.

%H Vladimir Pletser, <a href="/A229194/b229194.txt">Table of n, a(n) for n = 0..1000</a>

%H R. K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20.

%H I. Stewart, <a href="http://www.whydomath.org/Reading_Room_Material/ian_stewart/9505.html">Fibonacci Forgeries</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StrongLawofSmallNumbers.html">Strong Law of Small Numbers.</a>

%F a(n) = round(2^((n-3)/2) + 3^((n-3)/2)).

%p seq(round(2^((n-3)/2)+3^((n-3)/2)), n=0..50);

%t Table[Round[2^((n - 3)/2) + 3^((n - 3)/2)], {n, 0, 50}] (* _Vincenzo Librandi_, Sep 20 2013 *)

%o (Magma) [Round(2^((n-3)/2) + 3^((n-3)/2)): n in [0..50]]; // _Vincenzo Librandi_, Sep 20 2013

%Y Cf. A000045, A005181.

%K nonn

%O 0,4

%A _Vladimir Pletser_, Sep 15 2013