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

3,1

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.

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A022420 A326741 A034811 * A341770 A287167 A253975

Adjacent sequences: A283845 A283846 A283847 * A283849 A283850 A283851

Keyword

nonn

Author

N. J. A. Sloane, Apr 01 2017

Extensions

Edited and more terms by Robert Israel, Aug 23 2018

Status

approved