A283848 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

8, 23, 32, 86, 128, 339, 512, 1332, 2048, 5298, 8192, 21066, 32768, 83987, 131072, 334966, 524288, 1336988, 2097152, 5338206, 8388608, 21321234, 33554432, 85176636, 134217728, 340338398, 536870912, 1360073016, 2147483648, 5435820051, 8589934592, 21727481616, 34359738368, 86853790498, 137438953472

Offset: 3

Author

N. J. A. Sloane, Apr 01 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 3..3318
• 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 (2*n)^(-1)*add(phi(d)*4^(n/d),d=select(type,divisors(n),even))+5*2^(n-2)
  else 2^n
  fi
end proc:
map(f, [$1..40]); # Robert Israel, Aug 23 2018

Mathematica

a[n_] := If[EvenQ[n], (2n)^(-1) Sum[EulerPhi[d] 4^(n/d), {d, Select[ Divisors[n], EvenQ]}] + 5*2^(n-2), 2^n];
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Mar 23 2019, after Robert Israel *)

PARI

a(n) = if (n%2, 2^n, (2*n)^(-1)*sumdiv(n, d, if (!(d%2), eulerphi(d)*4^(n/d))) + 5*2^(n-2)); \\ Michel Marcus, Mar 23 2019

Formula

If n is even, a(n) = (2*n)^(-1)*Sum_{d|n, d even} phi(d)*4^(n/d) + 5*2^(n-2). - Robert Israel, Aug 23 2018 after Fujita (2017), Eq. (101) (set n=2, m=n).

If n is odd, a(n) = 2^n. For the even bisection see A284711.

Extensions

Edited and more terms by Robert Israel, Aug 23 2018