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

0,2

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.

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).

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

Example

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.

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 *)

Prog

(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
(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

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).

Sequence in context: A218522 A303072 A231839 * A164906 A141125 A164111

Adjacent sequences: A056447 A056448 A056449 * A056451 A056452 A056453

Keyword

nonn,easy

Author

Marks R. Nester

Extensions

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

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

Status

approved