A283847 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

2, 8, 36, 140, 522, 1920, 7030, 25704, 94302, 347488, 1286460, 4785300, 17879352, 67076096, 252579600, 954306220, 3616552422, 13743371072, 52356648380, 199909107900, 764873459802, 2932022620160, 11258982291252, 43303809016440, 166799919094902, 643371241120928

Offset: 3

Author

N. J. A. Sloane, Apr 01 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 3..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.

Crossrefs

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

Programs

Maple

f:= proc(n) uses numtheory;
  if n::even then (4*n)^(-1)*add(phi(d)*4^(n/d), d = select(type,divisors(n),odd)) - 2^(n-1)
  else (4*n)^(-1)*add(phi(d)*4^(n/d), d = divisors(n)) - 2^(n-1)
  fi
end proc:
map(f, [$3..50]); # Robert Israel, Aug 23 2018

Mathematica

a[n_] := If[EvenQ[n], (4n)^(-1) Sum[EulerPhi[d] 4^(n/d), {d, Select[ Divisors[n], OddQ]}] - 2^(n-1), (4n)^(-1) Sum[EulerPhi[d] 4^(n/d), {d, Divisors[n]}] - 2^(n-1)];
Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Mar 23 2019, after Robert Israel *)

Formula

a(n) = (4*n)^(-1)*(Sum_{d|n, d odd} phi(d)*4^(n/d)) - 2^(n-1). - Robert Israel, Aug 23 2018 after Fujita (2017), Eq. (100) (set n=2, m=n)