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A283846
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Number of n-gonal inositol homologs with 2 kinds of achiral proligands.
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9
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2, 6, 10, 31, 68, 226, 650, 2259, 7542, 27036, 96350, 352786, 1294652, 4806366, 17912120, 67160083, 252710672, 954641186, 3617076710, 13744708060, 52358745532, 199914446106, 764881848410, 2932043941394, 11259015845684, 43303894193076, 166800053312630
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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From Robert Israel, Aug 21 2018 after Fujita (2017), Eq. (99)(set n=2, m=n): (Start)
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).
if n is odd, a(n) = (4*n)^(-1)*Sum_{d|n} (phi(d)*4^(n/d)+2^(n-1). (End)
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MAPLE
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f:= proc(m) uses numtheory;
if m::even then 1/(4*m)*add(phi(d)*4^(m/d)*`if`(d::even, 2, 1), d = divisors(m))
+ 3*2^(m-2)
else
1/(4*m)*add(phi(d)*4^(m/d), d=divisors(m))+2^(m-1)
fi
end proc:
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MATHEMATICA
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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)];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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