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Number of n-gonal inositol homologs with 2 kinds of achiral proligands.
9

%I #17 Mar 23 2019 12:43:21

%S 2,8,36,140,522,1920,7030,25704,94302,347488,1286460,4785300,17879352,

%T 67076096,252579600,954306220,3616552422,13743371072,52356648380,

%U 199909107900,764873459802,2932022620160,11258982291252

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

%H Robert Israel, <a href="/A283847/b283847.txt">Table of n, a(n) for n = 3..1665</a>

%H Shinsaku Fujita, <a href="https://doi.org/10.1246/bcsj.20160369">alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method</a>, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.

%F From _Robert Israel_, Aug 23 2018 after Fujita (2017), Eq. (100)(set n=2, m=n): (Start)

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

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

%p f:= proc(n) uses numtheory;

%p if n::even then (4*n)^(-1)*add(phi(d)*4^(n/d), d = select(type,divisors(n),odd)) - 2^(n-1)

%p else (4*n)^(-1)*add(phi(d)*4^(n/d), d = divisors(n)) - 2^(n-1)

%p fi

%p end proc:

%p map(f, [$3..50]); # _Robert Israel_, Aug 23 2018

%t 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)];

%t Table[a[n], {n, 3, 50}] (* _Jean-François Alcover_, Mar 23 2019, after _Robert Israel_ *)

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

%K nonn

%O 3,1

%A _N. J. A. Sloane_, Apr 01 2017