A056452 a(n) = 6^floor((n+1)/2).

1, 6, 6, 36, 36, 216, 216, 1296, 1296, 7776, 7776, 46656, 46656, 279936, 279936, 1679616, 1679616, 10077696, 10077696, 60466176, 60466176, 362797056, 362797056, 2176782336, 2176782336, 13060694016, 13060694016, 78364164096

Offset: 0

Author

Marks R. Nester

Keywords

• nonn
• easy

Comments

Number of achiral rows of length n using up to six different colors. For a(3) = 36, the rows are AAA, ABA, ACA, ADA, AEA, AFA, BAB, BBB, BCB, BDB, BEB, BFB, CAC, CBC, CCC, CDC, CEC, CFC, DAD, DBD, DCD, DDD, DED, DFD, EAE, EBE, ECE, EDE, EEE, EFE, FAF, FBF, FCF, FDF, FEF, and FFF. - Robert A. Russell, Nov 08 2018

Also: a(n) is the number of palindromes with n digits using a maximum of six different symbols. - David A. Corneth, Nov 09 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 (0,6).

Crossrefs

Column k=6 of A321391.

Cf. A016116.

Cf. A000400 (oriented), A056308 (unoriented), A320524 (chiral).

Programs

Magma

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

Maple

A056452:=n->6^floor((n+1)/2);

Mathematica

Riffle[6^Range[0, 20], 6^Range[20]] (* Harvey P. Dale, Jun 18 2017 *)
Table[6^Ceiling[n/2], {n,0,40}] (* or *)
LinearRecurrence[{0, 6}, {1, 6}, 40] (* Robert A. Russell, Nov 08 2018 *)

Formula

a(n) = 6^floor((n+1)/2).

a(n) = 6*a(n-2). - Colin Barker, May 06 2012

G.f.: (1+6*x) / (1-6*x^2). - Colin Barker, May 06 2012 [Adapted to offset 0 by Robert A. Russell, Nov 08 2018]

a(n) = C(6,0)*A000007(n) + C(6,1)*A057427(n) + C(6,2)*A056453(n) + C(6,3)*A056454(n) + C(6,4)*A056455(n) + C(6,5)*A056456(n) + C(6,6)*A056457(n). - Robert A. Russell, Nov 08 2018

Extensions

a(0)=1 prepended by Robert A. Russell, Nov 08 2018

Name corrected by David A. Corneth, Nov 08 2018