A056450 a(n) = (3*2^n - (-2)^n)/2.

1, 4, 4, 16, 16, 64, 64, 256, 256, 1024, 1024, 4096, 4096, 16384, 16384, 65536, 65536, 262144, 262144, 1048576, 1048576, 4194304, 4194304, 16777216, 16777216, 67108864, 67108864, 268435456, 268435456, 1073741824, 1073741824, 4294967296

Offset: 0

Author

Marks R. Nester

Keywords

• nonn
• easy

Comments

Number of palindromes of length n using a maximum of four different symbols.

Number of achiral rows of n colors using up to four colors. - Robert A. Russell, Nov 09 2018

Interleaving of A000302 and 4*A000302.

Unsigned version of A141125.

Binomial transform is A164907. Second binomial transform is A164908. Third binomial transform is A057651. Fourth binomial transform is A016129.

Examples

			At length n=1 there are a(1)=4 palindromes, A, B, C, D.
At length n=2, there are a(2)=4 palindromes, AA, BB, CC, DD.
At length n=3, there are a(3)=16 palindromes, AAA, BBB, CCC, DDD, ABA, BAB, ... , CDC, DCD.
		

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
• Sean A. Irvine, Walks on Graphs.
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (0,4).

Crossrefs

Column k=4 of A321391.

Cf. A000302 (powers of 4), A056450, A141125, A164907, A164908, A057651, A016129.

Cf. A016116.

Essentially the same as A213173.

Cf. A000302 (oriented), A032121 (unoriented), A032087(n>1) (chiral).

Programs

Magma

[ (3*2^n-(-2)^n)/2: n in [0..31] ];

Magma

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

Mathematica

Table[4^Ceiling[n/2], {n,0,40}] (* or *)
CoefficientList[Series[(1 + 4 x)/((1 + 2 x) (1 - 2 x)), {x, 0, 31}], x] (* or *)
LinearRecurrence[{0, 4}, {1, 4}, 40] (* Robert A. Russell, Nov 07 2018 *)

PARI

a(n)=4^((n+1)\2) \\ Charles R Greathouse IV, Apr 08 2012

PARI

a(n)=(3*2^n-(-2)^n)/2 \\ Charles R Greathouse IV, Oct 03 2016

Formula

a(n) = 4^floor((n+1)/2).

a(n) = 4*a(n-2) for n > 1; a(0) = 1, a(1) = 4.

G.f.: (1+4*x) / (1-4*x^2). - R. J. Mathar, Jan 19 2011 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]

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

a(n) = 4*abs(A164111(n-1)). - R. J. Mathar, Jan 19 2011

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

Extensions

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

Edited by N. J. A. Sloane, Sep 29 2019