

A056503


Number of periodic palindromic structures of length n using a maximum of two different symbols.


6



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

1,2


COMMENTS

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


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

Andrew Howroyd, Table of n, a(n) for n = 1..200


FORMULA

a(2n+1) = A164090(2n+1)/2 = 2^n, a(2n) = (A164090(2n) + A045674(n))/2.  Andrew Howroyd, Sep 29 2017


MATHEMATICA

(* b = A164090, c = A045674 *)
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/21) + c[n/2], 2^((n1)/2)];
a[n_?OddQ] := b[n]/2; a[n_?EvenQ] := (1/2)*(b[n] + c[n/2]);
Array[a, 45] (* JeanFrançois Alcover, Oct 08 2017, after Andrew Howroyd *)


CROSSREFS

Cf. A016116, A045674, A056508, A164090, A285012.
Sequence in context: A222738 A005308 A151532 * A256217 A055636 A206559
Adjacent sequences: A056500 A056501 A056502 * A056504 A056505 A056506


KEYWORD

nonn,changed


AUTHOR

Marks R. Nester


EXTENSIONS

a(17)a(45) from Andrew Howroyd, Apr 07 2017


STATUS

approved



