A283846 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

2, 6, 10, 31, 68, 226, 650, 2259, 7542, 27036, 96350, 352786, 1294652, 4806366, 17912120, 67160083, 252710672, 954641186, 3617076710, 13744708060, 52358745532, 199914446106, 764881848410, 2932043941394, 11259015845684, 43303894193076, 166800053312630

Offset: 1

Author

N. J. A. Sloane, Apr 01 2017

Keywords

• nonn

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.

Crossrefs

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

Programs

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 *)

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) = 2^(n-1) + (4*n)^(-1)*Sum_{d|n} phi(d)*4^(n/d). (End)

Extensions

a(1)-a(2) prepended by Travis Scott, Jan 09 2023