|
%I #19 Mar 21 2023 18:07:06
%S 0,1,2,2,3,3,3,4,4,3,5,5,3,5,6,4,5,5,5,7,5,3,7,8,4,5,7,5,7,7,5,7,5,5,
%T 10,8,3,5,9,7,7,7,5,9,7,3,9,10,6,7,7,5,7,9,9,9,5,3,11,11,3,7,10,8,9,7,
%U 5,7,9,7,11,11,3,7,9,7,9,7,9,12,6,3,11,13
%N a(n) is the number of dihedral symmetries D_{2m} (m >= 3) that configurations of n non-overlapping equal circles can possess.
%F For n >= 2, a(n) = A274010(n) - 1 = A023645(n) + A023645(n-1) = tau(n) + tau(n-1) - 3, where tau(n) = A000005(n), the number of divisors of n.
%e a(2) = 0 because the configuration of 2 circles only possesses D_{4} symmetry.
%e a(6) = 3 because configurations of 6 circles can have three dihedral symmetries: D_{12} (6 circles arranged in regular hexagon shape), D_{10} (5 circles arranged in regular pentagon shape and the other circle in the center of the pentagon), and D_{6} (6 circles arranged in equilateral triangle shape).
%t Table[DivisorSigma[0,n]+DivisorSigma[0,n-1]-3,{n,2,85}] (* _Stefano Spezia_, Apr 06 2021 *)
%o (Python)
%o from sympy import divisor_count
%o for n in range(2, 101):
%o print(divisor_count(n) + divisor_count(n - 1) - 3, end=", ")
%Y Cf. A000005, A023645, A274010.
%K nonn
%O 2,3
%A _Ya-Ping Lu_, Apr 02 2021
|
