A108411 a(n) = 3^floor(n/2). Powers of 3 repeated.

1, 1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 0

Author

Ralf Stephan, Jun 05 2005

Keywords

• nonn
• easy

Comments

a(n) is the Parker sequence for the automorphism group of the limit of the class of oriented graphs; a(n) counts the finite circulant structures in that class. - N-E. Fahssi, Feb 18 2008

Complete sequence: every positive integer is the sum of members of this sequence. - Charles R Greathouse IV, Jul 19 2012

Conjecture: a(n+1) is the number of distinct subsets S of {0,1,2,...,n} such that the sumset S+S does not contain n. - Michael Chu, Oct 05 2021. Andrew Howroyd, Nov 20 2021: The conjecture is true: If there are m pairs of numbers that add to n then inclusion/exclusion gives sum(k=0, m, binomial(m,k)*(-1)^k*2^(2*m-2*k)) as the number of sets that don't contain any of those pairs which equals 3^m. For even n , n/2 cannot be included in any set.

Also, number of walks of length n in the graph K_{1,3} (the graph with edges {1,2}, {1,3}, {1,4}) starting at one of the degree 1 vertices. - Sean A. Irvine, May 30 2025

Examples

			a(6) = 27; 3^floor(6/2) = 3^floor(3) = 3^3 = 27.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..3000
• D. A. Gewurz and F. Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., 6 (2003), 03.1.6.
• Index entries for linear recurrences with constant coefficients, signature (0,3).

Crossrefs

Essentially the same as A056449 and A162436.

Cf. A000244, A004526, A016116, A152815, A158020.

Programs

Haskell

a108411 = (3 ^) . flip div 2  -- Reinhard Zumkeller, May 01 2014

Magma

[3^Floor(n/2): n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

Maple

A108411:=n->3^floor(n/2); seq(A108411(k), k=0..100); # Wesley Ivan Hurt, Nov 01 2013

Mathematica

Table[3^Floor[n/2], {n,0,100}] (* Wesley Ivan Hurt, Nov 01 2013 *)

PARI

a(n)=3^floor(n/2);

Python

def A108411(n): return 3**(n>>1) # Chai Wah Wu, Oct 28 2024

Formula

O.g.f.: (1+x)/(1-3*x^2). - R. J. Mathar, Apr 01 2008

a(n) = 3^(n/2)*((1+(-1)^n)/2+(1-(-1)^n)/(2*sqrt(3))). - Paul Barry, Nov 12 2009

a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011

a(n) = (-1)^n*sum(A158020(n,k)*2^k, 0<=k<=n). - Philippe Deléham, Dec 01 2011

a(n) = sum(A152815(n,k)*2^k, 0<=k<=n). - Philippe Deléham, Apr 22 2013

a(n) = 3^A004526(n). - Michel Marcus, Aug 30 2014

E.g.f.: cosh(sqrt(3)*x) + sinh(sqrt(3)*x)/sqrt(3). - Stefano Spezia, Dec 31 2022

Extensions

Incorrect formula removed by Michel Marcus, Oct 06 2021