OFFSET
1,1
COMMENTS
Counts A032275 up to paired color permutation (equivalent to full color permutation on the 2-tuples of two subcolors, e.g., convert quaternary beads 0 1 2 3 to dibit beads 00 01 10 11). - Travis Scott, Jan 09 2023
LINKS
Robert Israel, Table of n, a(n) for n = 1..1665
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
Yi Hu, Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models, Master's Thesis, Duke Univ. (2021).
Yi Hu and Patrick Charbonneau, Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices, arXiv:2106.08442 [cond-mat.stat-mech], 2021.
FORMULA
From Robert Israel, Aug 21 2018 after Fujita (2017), Eq. (99)(set n=2, m=n): (Start)
if n is even, a(n) = (4*n)^(-1)*(Sum_{d|n} phi(d)*4^(n/d) + Sum_{d|n, d even} phi(d)*4^(n/d)) + 3*2^(n-2).
if n is odd, a(n) = (4*n)^(-1)*Sum_{d|n} (phi(d)*4^(n/d)+2^(n-1). (End)
MAPLE
f:= proc(m) uses numtheory;
if m::even then 1/(4*m)*add(phi(d)*4^(m/d)*`if`(d::even, 2, 1), d = divisors(m))
+ 3*2^(m-2)
else
1/(4*m)*add(phi(d)*4^(m/d), d=divisors(m))+2^(m-1)
fi
end proc:
map(f, [$1..100]); # Robert Israel, Aug 21 2018
MATHEMATICA
f[m_] := If[EvenQ[m], 1/(4m)*Sum[EulerPhi[d]*4^(m/d)*If[EvenQ[d], 2, 1], {d, Divisors[m]}]+ 3*2^(m-2), 1/(4m)*Sum[EulerPhi[d]*4^(m/d), {d, Divisors[m]}] + 2^(m-1)];
f /@ Range[1, 25] (* Jean-François Alcover, Feb 26 2019, after Robert Israel *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 01 2017
EXTENSIONS
a(1)-a(2) prepended by Travis Scott, Jan 09 2023
STATUS
approved