login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A056503 Number of periodic palindromic structures of length n using a maximum of two different symbols. 8
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

Number of cyclic compositions (necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018

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

EXAMPLE

From Gus Wiseman, Sep 16 2018: (Start)

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)

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 *)

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 *)

CROSSREFS

Row sums of A179181.

Cf. A016116, A045674, A056508, A164090, A285012.

Cf. A000740, A000837, A008965, A025065, A059966, A242414, A296302, A317085, A317086, A317087, A318731.

Sequence in context: A222738 A005308 A151532 * A256217 A055636 A206559

Adjacent sequences:  A056500 A056501 A056502 * A056504 A056505 A056506

KEYWORD

nonn

AUTHOR

Marks R. Nester

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 29 05:25 EST 2020. Contains 332353 sequences. (Running on oeis4.)