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A283846 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

%I A283846 #55 Mar 08 2025 18:38:57
%S A283846 2,6,10,31,68,226,650,2259,7542,27036,96350,352786,1294652,4806366,
%T A283846 17912120,67160083,252710672,954641186,3617076710,13744708060,
%U A283846 52358745532,199914446106,764881848410,2932043941394,11259015845684,43303894193076,166800053312630
%N A283846 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.
%C A283846 Counts A032275 up to paired color permutation (equivalent to full color permutation on the 2-tuples of two subcolors, e.g., convert quaternary beads 0 1 2 3 to dibit beads 00 01 10 11). - _Travis Scott_, Jan 09 2023
%H A283846 Robert Israel, <a href="/A283846/b283846.txt">Table of n, a(n) for n = 1..1665</a>
%H A283846 Shinsaku Fujita, <a href="https://www.webfile.jp/lite-tcsj/dl.php?i=2071&amp;s=aaa3068608b595219b60">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.
%H A283846 Yi Hu, <a href="https://hdl.handle.net/10161/23828">Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models</a>, Master's Thesis, Duke Univ. (2021).
%H A283846 Yi Hu and Patrick Charbonneau, <a href="https://arxiv.org/abs/2106.08442">Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices</a>, arXiv:2106.08442 [cond-mat.stat-mech], 2021.
%F A283846 From _Robert Israel_, Aug 21 2018 after Fujita (2017), Eq. (99)(set n=2, m=n): (Start)
%F A283846 if n is even, a(n) = (4*n)^(-1)*(Sum_{d|n} phi(d)*4^(n/d) + Sum_{d|n, d even} phi(d)*4^(n/d)) + 3*2^(n-2).
%F A283846 if n is odd, a(n) = 2^(n-1) + (4*n)^(-1)*Sum_{d|n} phi(d)*4^(n/d). (End)
%p A283846 f:=  proc(m) uses numtheory;
%p A283846   if m::even then 1/(4*m)*add(phi(d)*4^(m/d)*`if`(d::even,2,1), d = divisors(m))
%p A283846 + 3*2^(m-2)
%p A283846   else
%p A283846 1/(4*m)*add(phi(d)*4^(m/d),d=divisors(m))+2^(m-1)
%p A283846   fi
%p A283846 end proc:
%p A283846 map(f, [$1..100]); # _Robert Israel_, Aug 21 2018
%t A283846 f[m_] := If[EvenQ[m], 1/(4m)*Sum[EulerPhi[d]*4^(m/d)*If[EvenQ[d], 2, 1], {d, Divisors[m]}]+ 3*2^(m-2), 1/(4m)*Sum[EulerPhi[d]*4^(m/d), {d, Divisors[m]}] + 2^(m-1)];
%t A283846 f /@ Range[1, 25] (* _Jean-François Alcover_, Feb 26 2019, after _Robert Israel_ *)
%Y A283846 The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.
%K A283846 nonn
%O A283846 1,1
%A A283846 _N. J. A. Sloane_, Apr 01 2017
%E A283846 a(1)-a(2) prepended by _Travis Scott_, Jan 09 2023