A056449 a(n) = 3^floor((n+1)/2).

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

Marks R. Nester

Keywords

• nonn
• easy

Comments

One followed by powers of 3 with positive exponent, repeated. - Omar E. Pol, Jul 27 2009

Number of achiral rows of n colors using up to three colors. E.g., for a(3) = 9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC. - Robert A. Russell, Nov 07 2018

References

• M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (0,3).

Crossrefs

Column k=3 of A321391.

Cf. A000007, A000244, A016116, A056391, A056453, A056454, A057427.

Essentially the same as A108411 and A162436.

Cf. A000244 (oriented), A032120 (unoriented), A032086(n>1) (chiral).

Programs

Magma

[3^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Mathematica

Riffle[3^Range[0, 20], 3^Range[20]] (* Harvey P. Dale, Jan 21 2015 *)
Table[3^Ceiling[n/2], {n,0,40}] (* or *)
LinearRecurrence[{0, 3}, {1, 3}, 40] (* Robert A. Russell, Nov 07 2018 *)

PARI

a(n)=3^floor((n+1)/2); \\ Joerg Arndt, Apr 23 2013

Python

def A056449(n): return 3**(n+1>>1) # Chai Wah Wu, Oct 28 2024

Formula

G.f.: (1 + 3*x) / (1 - 3*x^2). - R. J. Mathar, Jul 06 2011 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]

a(n) = k^ceiling(n/2), where k = 3 is the number of possible colors. - Robert A. Russell, Nov 07 2018

a(n) = C(3,0)*A000007(n) + C(3,1)*A057427(n) + C(3,2)*A056453(n) + C(3,3)*A056454(n). - Robert A. Russell, Nov 08 2018

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

Extensions

Edited by N. J. A. Sloane at the suggestion of Klaus Brockhaus, Jul 03 2009

a(0)=1 prepended by Robert A. Russell, Nov 07 2018