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

Offset

3,1

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.

Formula

From Robert Israel, Aug 23 2018 after Fujita (2017), Eq. (100)(set n=2, m=n): (Start)

if n is even, a(n) = (4*n)^(-1)*(Sum_{d|n, d odd} phi(d)*4^(n/d) - 2^(n-1).

if n is odd, a(n) = (4*n)^(-1)*Sum_{d|n} phi(d)*4^(n/d) - 2^(n-1). (End)

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 *)

Crossrefs

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

Sequence in context: A248861 A323677 A295501 * A123290 A321110 A228791

Adjacent sequences: A283844 A283845 A283846 * A283848 A283849 A283850

Keyword

nonn

Author

N. J. A. Sloane, Apr 01 2017

Status

approved