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A307813
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a(n) = (5/32)*4^n - floor((n^2 + 1)/2)*2^(n - 2).
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0
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0, 8, 56, 352, 1760, 8192, 35712, 151040, 624128, 2547712, 10311680, 41541632, 166846464, 668991488, 2679603200, 10726801408, 42925948928, 171746263040, 687078899712, 2748525314048
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OFFSET
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3,2
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COMMENTS
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This was formerly conjectured (by Erdos and Guy, 1973) to be the crossing number of the n-hypercube graph for n >= 3; cr(Q_n) = 0 for n = 0, 1, 2, 3.
Embeddings of Q_n with cr(Q_n) = a(n) are known for n = 5 and 6. However, the Erdos-Guy conjecture is now refuted since cr(Q_7) <= 1744 < a(7) = 1760 (Clancy et al. 2019).
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LINKS
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FORMULA
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a(n) = 5/32*4^n - floor((n^2 + 1)/2)*2^(n - 2).
a(n) = 2^(n - 5)*(5*2^n - 4*n^2 + 2 (-1)^n - 2).
a(n) = 8*a(n-1) - 16*a(n-2) - 16*a(n-3) + 80*a(n-4) - 64*a(n-5).
G.f.: -8*x^4*(-1 + x - 4*x^2 + 4*x^3)/((-1 + 2*x)^3*(-1 + 2*x + 8*x^2)).
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MATHEMATICA
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Table[5/32 4^n - Floor[(n^2 + 1)/2] 2^(n - 2), {n, 3, 20}]
Table[2^(n - 5) (5 2^n - 4 n^2 + 2 (-1)^n - 2), {n, 3, 20}]
LinearRecurrence[{8, -16, -16, 80, -64}, {0, 8, 56, 352, 1760}, 20]
CoefficientList[Series[-8 x (-1 + x - 4 x^2 + 4 x^3)/((-1 + 2 x)^3 (-1 + 2 x + 8 x^2)), {x, 0, 20}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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