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A056450
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a(n) = (3*2^n - (-2)^n)/2.
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25
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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
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listen;
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internal format)
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OFFSET
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0,2
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COMMENTS
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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
Binomial transform is A164907. Second binomial transform is A164908. Third binomial transform is A057651. Fourth binomial transform is A016129.
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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^floor((n+1)/2).
a(n) = 4*a(n-2) for n > 1; a(0) = 1, a(1) = 4.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(Magma) [ (3*2^n-(-2)^n)/2: n in [0..31] ];
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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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