cp's OEIS Frontend

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

%I A056476 #26 Feb 18 2025 17:36:17
%S A056476 1,1,0,1,1,3,2,7,6,14,12,31,27,63,56,123,120,255,238,511,495,1015,992,
%T A056476 2047,2010,4092,4032,8176,8127,16383,16242,32767,32640,65503,65280,
%U A056476 131061,130788,262143,261632,524223,523770,1048575,1047494,2097151,2096127,4194162
%N A056476 Number of primitive (aperiodic) palindromic structures of length n using a maximum of two different symbols.
%C A056476 Permuting the symbols will not change the structure.
%C A056476 a(n) = A056481(n) for n > 1. - _Jonathan Frech_, May 21 2021
%D A056476 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]
%H A056476 Michael De Vlieger, <a href="/A056476/b056476.txt">Table of n, a(n) for n = 0..5000</a>
%F A056476 a(n) = Sum_{d|n} mu(d)*A016116(n/d-1) for n > 0.
%F A056476 a(n) = Sum_{k=1..2} A284826(n, k) for n > 0. - _Andrew Howroyd_, May 21 2021
%F A056476 a(n) = A056458(n)/2 for n>=1. - _Alois P. Heinz_, Feb 18 2025
%e A056476 Example from _Jonathan Frech_, May 21 2021: (Start)
%e A056476 The a(9)=14 lexicographically earliest equivalence class members in the alphabet {0,1} are:
%e A056476   000010000
%e A056476   000101000
%e A056476   000111000
%e A056476   001000100
%e A056476   001010100
%e A056476   001101100
%e A056476   001111100
%e A056476   010000010
%e A056476   010101010
%e A056476   010111010
%e A056476   011000110
%e A056476   011010110
%e A056476   011101110
%e A056476   011111110
%e A056476 (End)
%t A056476 Table[DivisorSum[n, MoebiusMu[#]*2^Floor[(n/# - 1)/2] &], {n, 46}] (* _Michael De Vlieger_, May 21 2021 *)
%o A056476 (PARI) a(n) = if(n==0, 1, sumdiv(n, d, moebius(d)*2^((n/d-1)\2))) \\ _Andrew Howroyd_, May 21 2021
%o A056476 (Python)
%o A056476 from sympy import mobius, divisors
%o A056476 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
%Y A056476 Cf. A056458, A056481, A284826.
%K A056476 nonn
%O A056476 0,6
%A A056476 _Marks R. Nester_
%E A056476 Definition clarified by _Jonathan Frech_, May 21 2021
%E A056476 a(0)=1 prepended and a(32)-a(45) from _Andrew Howroyd_, May 21 2021