A056476 Number of primitive (aperiodic) palindromic structures of length n using a maximum of two different symbols.

1, 1, 0, 1, 1, 3, 2, 7, 6, 14, 12, 31, 27, 63, 56, 123, 120, 255, 238, 511, 495, 1015, 992, 2047, 2010, 4092, 4032, 8176, 8127, 16383, 16242, 32767, 32640, 65503, 65280, 131061, 130788, 262143, 261632, 524223, 523770, 1048575, 1047494, 2097151, 2096127, 4194162

Offset: 0

Author

Marks R. Nester

Keywords

• nonn

Comments

Permuting the symbols will not change the structure.

a(n) = A056481(n) for n > 1. - Jonathan Frech, May 21 2021

Examples

			Example from _Jonathan Frech_, May 21 2021: (Start)
The a(9)=14 lexicographically earliest equivalence class members in the alphabet {0,1} are:
  000010000
  000101000
  000111000
  001000100
  001010100
  001101100
  001111100
  010000010
  010101010
  010111010
  011000110
  011010110
  011101110
  011111110
(End)
		

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

• Michael De Vlieger, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A056458, A056481, A284826.

Programs

Mathematica

Table[DivisorSum[n, MoebiusMu[#]*2^Floor[(n/# - 1)/2] &], {n, 46}] (* Michael De Vlieger, May 21 2021 *)

PARI

a(n) = if(n==0, 1, sumdiv(n, d, moebius(d)*2^((n/d-1)\2))) \\ Andrew Howroyd, May 21 2021

Python

from sympy import mobius, divisors
def A056476(n): return sum(mobius(n//d)<<(d-1>>1) for d in divisors(n, generator=True)) if n else 1 # Chai Wah Wu, Feb 18 2024

Formula

a(n) = Sum_{d|n} mu(d)*A016116(n/d-1) for n > 0.

a(n) = Sum_{k=1..2} A284826(n, k) for n > 0. - Andrew Howroyd, May 21 2021

a(n) = A056458(n)/2 for n>=1. - Alois P. Heinz, Feb 18 2025

Extensions

Definition clarified by Jonathan Frech, May 21 2021

a(0)=1 prepended and a(32)-a(45) from Andrew Howroyd, May 21 2021