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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 (list; graph; refs; listen; history; text; internal format)
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/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]);

Array[a, 45] (* Jean-Fran├žois 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

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Last modified October 17 18:14 EDT 2017. Contains 293471 sequences.