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A056508 Number of periodic palindromic structures of length n using exactly two different symbols. 3
0, 1, 1, 3, 3, 6, 7, 13, 15, 25, 31, 50, 63, 99, 127, 197, 255, 391, 511, 777, 1023, 1551, 2047, 3090, 4095, 6175, 8191, 12323, 16383, 24639, 32767, 49221, 65535, 98431, 131071, 196743, 262143, 393471, 524287, 786697, 1048575, 1573375, 2097151, 3146255, 4194303 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

For odd n, a palindrome cannot be the complement of itself, so a(n) is given by A284855(n,2)/2 - 1. - Andrew Howroyd, Apr 08 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

Giovanni Resta, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A056503(n) - 1.

a(2n + 1) = 2^n - 1. - Andrew Howroyd, Apr 07 2017

EXAMPLE

From Andrew Howroyd, Apr 07 2017: (Start)

Example for n=6:

Periodic symmetry means results are either in the form abccba or abcdcb.

There are 3 binary words in the form abccba that start with 0 and contain a 1 which are 001100, 010010, 011110. Of these, 011110 is equivalent to 001100 after rotation.

There are 7 binary words in the form abcdcb that start with 0 and contain a 1 which are 000100, 001010, 001110, 010001, 010101, 011011, 011111. Of these, 011111 is equivalent to 000100, 010001 is equivalent to 001010 and 011011 is equivalent to 010010 from the first set.

There are therefore a total of 7 + 3 - 4 = 6 equivalence classes so a(6) = 6.

(End)

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_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])] - 1;

Array[a, 45] (* Jean-Fran├žois Alcover, Jun 29 2018, after Andrew Howroyd *)

CROSSREFS

Column 2 of A285012.

Cf. A052551.

Sequence in context: A325834 A241832 A027187 * A050065 A298732 A078477

Adjacent sequences:  A056505 A056506 A056507 * A056509 A056510 A056511

KEYWORD

nonn

AUTHOR

Marks R. Nester

EXTENSIONS

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

STATUS

approved

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Last modified November 12 19:41 EST 2019. Contains 329078 sequences. (Running on oeis4.)