|
%I #24 Feb 19 2024 01:48:56
%S 1,1,0,1,1,3,2,7,6,14,12,31,27,63,56,123,120,255,238,511,495,1015,992,
%T 2047,2010,4092,4032,8176,8127,16383,16242,32767,32640,65503,65280,
%U 131061,130788,262143,261632,524223,523770,1048575,1047494,2097151,2096127,4194162
%N Number of primitive (aperiodic) palindromic structures of length n using a maximum of two different symbols.
%C Permuting the symbols will not change the structure.
%C a(n) = A056481(n) for n > 1. - _Jonathan Frech_, May 21 2021
%D 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 Michael De Vlieger, <a href="/A056476/b056476.txt">Table of n, a(n) for n = 0..5000</a>
%F a(n) = Sum_{d|n} mu(d)*A016116(n/d-1) for n > 0.
%F a(n) = Sum_{k=1..2} A284826(n, k) for n > 0. - _Andrew Howroyd_, May 21 2021
%e Example from _Jonathan Frech_, May 21 2021: (Start)
%e The a(9)=14 lexicographically earliest equivalence class members in the alphabet {0,1} are:
%e 000010000
%e 000101000
%e 000111000
%e 001000100
%e 001010100
%e 001101100
%e 001111100
%e 010000010
%e 010101010
%e 010111010
%e 011000110
%e 011010110
%e 011101110
%e 011111110
%e (End)
%t Table[DivisorSum[n, MoebiusMu[#]*2^Floor[(n/# - 1)/2] &], {n, 46}] (* _Michael De Vlieger_, May 21 2021 *)
%o (PARI) a(n) = if(n==0, 1, sumdiv(n, d, moebius(d)*2^((n/d-1)\2))) \\ _Andrew Howroyd_, May 21 2021
%o (Python)
%o from sympy import mobius, divisors
%o 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 Cf. A056458, A056481, A284826.
%K nonn
%O 0,6
%A _Marks R. Nester_
%E Definition clarified by _Jonathan Frech_, May 21 2021
%E a(0)=1 prepended and a(32)-a(45) from _Andrew Howroyd_, May 21 2021
|
