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A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols. 5
1, 1, 1, 1, 2, 3, 4, 7, 10, 14, 21, 31, 42, 63, 91, 123, 184, 255, 371, 511, 750, 1015, 1519, 2047, 3030, 4092, 6111, 8176, 12222, 16383, 24486, 32767, 49024, 65503, 98175, 131061, 196308, 262143, 392959, 524223, 785910, 1048575, 1572256, 2097151, 3144702, 4194162 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

Number of Lyndon compositions (aperiodic 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 = 0..1000

FORMULA

a(n) = Sum_{d|n} mu(d)*A056503(n/d) for n > 0.

a(n) = Sum_{k=1..2} A285037(n, k). - Andrew Howroyd, Apr 08 2017

G.f.: 1 + (1/2)*Sum_{k>=1} mu(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - mu(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)). - Andrew Howroyd, Sep 27 2019

EXAMPLE

From Gus Wiseman, Sep 16 2018: (Start)

The sequence of palindromic Lyndon compositions begins:

  (1)  (2)  (3)  (4)    (5)    (6)      (7)

                 (112)  (113)  (114)    (115)

                        (122)  (1122)   (133)

                               (11112)  (223)

                                        (11113)

                                        (11212)

                                        (11122)

(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)];

a56503[n_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])];

a[n_] := DivisorSum[n, MoebiusMu[#] a56503[n/#]&];

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

PROG

(PARI) a(n) = {if(n < 1, n==0, sumdiv(n, d, moebius(d)*(2 + d%2)*(2^(n/d\2)))/(4 - n%2))} \\ Andrew Howroyd, Sep 26 2019

(PARI) seq(n) = Vec(1 + (1/2)*sum(k=1, n, moebius(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - moebius(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)) + O(x*x^n))) \\ Andrew Howroyd, Sep 27 2019

CROSSREFS

Row sums of A179317.

Cf. A008965, A025065, A056476, A059966, A242414, A285037, A317085, A317086, A317087, A318731.

Sequence in context: A184639 A035565 A240489 * A056518 A160644 A267459

Adjacent sequences:  A056510 A056511 A056512 * A056514 A056515 A056516

KEYWORD

nonn

AUTHOR

Marks R. Nester

EXTENSIONS

a(17)-a(45) from Andrew Howroyd, Apr 08 2017

a(0)=1 prepended by Andrew Howroyd, Sep 27 2019

STATUS

approved

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Last modified November 19 08:36 EST 2019. Contains 329318 sequences. (Running on oeis4.)