A056486 a(n) = (9*2^n + (-2)^n)/4 for n>0.

4, 10, 16, 40, 64, 160, 256, 640, 1024, 2560, 4096, 10240, 16384, 40960, 65536, 163840, 262144, 655360, 1048576, 2621440, 4194304, 10485760, 16777216, 41943040, 67108864, 167772160, 268435456, 671088640, 1073741824, 2684354560, 4294967296, 10737418240

Offset: 1

Author

Marks R. Nester

Keywords

• nonn
• easy

Comments

Old name was: "Number of periodic palindromes using a maximum of four different symbols".

Number of necklaces with n beads and 4 colors that are the same when turned over and hence have reflection symmetry. - Herbert Kociemba, Nov 24 2016

Examples

			G.f. = 4*x + 10*x^2 + 16*x^3 + 40*x^4 + 64*x^5 + 160*x^6 + 256*x^7 + 640*x^8 + ...
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
		

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 = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,4).

Crossrefs

Cf. A029744, A038754, A056450.

Column 4 of A284855.

Programs

Magma

[(9*2^n + (-2)^n)/4 : n in [1..50]]; // Wesley Ivan Hurt, Nov 24 2016

Maple

A056486:=n->(9*2^n + (-2)^n)/4: seq(A056486(n), n=1..50); # Wesley Ivan Hurt, Nov 24 2016

Mathematica

CoefficientList[Series[-1+(1+4*x+6*x^2)/(1-4*x^2),{x,0,30}],x] (* Herbert Kociemba, Nov 24 2016 *)
k=4; Table[(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2]) / 2, {n, 1, 30}] (* Robert A. Russell, Sep 21 2018 *)

PARI

a(n) = (9*2^n+(-2)^n)/4; \\ Altug Alkan, Sep 21 2018

SageMath

[2^(n-2)*(9+(-1)^n) for n in range(1,51)] # G. C. Greubel, Mar 23 2024

Formula

a(n) = 4^((n+1)/2) for n odd, a(n) = 4^(n/2)*5/2 for n even.

From Colin Barker, Jul 08 2012: (Start)

a(n) = 4*a(n-2).

G.f.: 2*x*(2+5*x)/((1-2*x)*(1+2*x)). (End)

G.f.: -1 + (1+4*x+6*x^2)/(1-4*x^2). - Herbert Kociemba, Nov 24 2016

E.g.f.: 5*sinh(x)^2 + 2*sinh(2*x). - Ilya Gutkovskiy, Nov 24 2016

a(n) = ( 4^floor((n+1)/2) + 4^ceiling((n+1)/2) )/2. - Robert A. Russell, Sep 21 2018

Extensions

Better name from Ralf Stephan, Jul 18 2013