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A056486
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a(n) = (9*2^n + (-2)^n)/4 for n>0.
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8
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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]
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LINKS
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FORMULA
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a(n) = 4^((n+1)/2) for n odd, a(n) = 4^(n/2)*5/2 for n even.
a(n) = 4*a(n-2).
G.f.: 2*x*(2+5*x)/((1-2*x)*(1+2*x)). (End)
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EXAMPLE
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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.
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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