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

0,6

Comments

Permuting the symbols will not change the structure.

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

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

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

Example

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)

Mathematica

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

Prog

(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

Crossrefs

Cf. A056458, A056481, A284826.

Sequence in context: A125718 A268821 A014841 * A056481 A366276 A269386

Adjacent sequences: A056473 A056474 A056475 * A056477 A056478 A056479

Keyword

nonn

Author

Marks R. Nester

Extensions

Definition clarified by Jonathan Frech, May 21 2021

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

Status

approved