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A056503
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Number of periodic palindromic structures of length n using a maximum of two different symbols.
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10
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1, 2, 2, 4, 4, 7, 8, 14, 16, 26, 32, 51, 64, 100, 128, 198, 256, 392, 512, 778, 1024, 1552, 2048, 3091, 4096, 6176, 8192, 12324, 16384, 24640, 32768, 49222, 65536, 98432, 131072, 196744, 262144, 393472, 524288, 786698, 1048576, 1573376, 2097152, 3146256, 4194304
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OFFSET
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1,2
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COMMENTS
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For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.
A periodic palindrome is just a necklace that is equivalent to its reverse. The number of binary periodic palindromes of length n is given by A164090(n). A binary periodic palindrome can only be equivalent to its complement when there are an equal number of 0's and 1's. - Andrew Howroyd, Sep 29 2017
Number of cyclic compositions (necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018
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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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EXAMPLE
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The sequence of palindromic cyclic compositions begins:
(1) (2) (3) (4) (5) (6) (7)
(11) (111) (22) (113) (33) (115)
(112) (122) (114) (133)
(1111) (11111) (222) (223)
(1122) (11113)
(11112) (11212)
(111111) (11122)
(1111111)
(End)
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MATHEMATICA
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b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1; c[n_] := c[n] = If[EvenQ[n], 2^(n/2-1) + c[n/2], 2^((n-1)/2)];
a[n_?OddQ] := b[n]/2; a[n_?EvenQ] := (1/2)*(b[n] + c[n/2]);
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Function[q, And[Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And], Array[SameQ[RotateRight[q, #], Reverse[RotateRight[q, #]]]&, Length[q], 1, Or]]]]], {n, 15}] (* Gus Wiseman, Sep 16 2018 *)
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CROSSREFS
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Cf. A000740, A000837, A008965, A025065, A059966, A242414, A296302, A317085, A317086, A317087, A318731.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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