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A295223
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Number of tilings of the n X n torus, using diagonal lines to connect the gridpoints.
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10
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1, 4, 18, 669, 170440, 238773358, 1436110601256, 36028800332480074, 3731252530927004638384, 1584563250286480205777197264, 2746338834266357074512496613490144, 19358285762613388151183577985346072926384, 553468075675608205710323628035216140349636855680
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OFFSET
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1,2
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LINKS
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EXAMPLE
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For n = 3, the following four tilings are considered equivalent:
*---*->-+---+ +---+->-*---* *---*->-+---+ +---+->-+---+
| / | \ | \ | | / | / | \ | | \ | / | / | | / | \ | \ |
*---*---+---+ +---+---*---* *---*---+---+ *---*---+---+
^ / | / | \ ^ = ^ / | \ | \ ^ = ^ \ | / | \ ^ = ^ \ | / | / ^
+---+---+---+ +---+---+---+ +---+---+---+ *---*---+---+
| \ | / | / | | \ | \ | / | | / | \ | \ | | \ | / | \ |
+---+->-+---+ +---+->-+---+ +---+->-+---+ +---+->-+---+
The transformations are horizontal reflection, shifting to the right, and shifting down.
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MATHEMATICA
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a[n_] := 1/(8*n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + If[OddQ[n], n^2*2^((n^2 + 1)/2), n^2/4 (3*2^(n^2/2) + 2^((n^2 + 4)/2))] + 2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[c] (2^(n^2/LCM[2, c]) + If[OddQ[c], 0, 2^(n^2/c)])]], n*DivisorSum[n, Function[c, EulerPhi[c]*If[OddQ[c], 0, 2^(n^2/c)]]]] + If[OddQ[n], 0, n^2 (2^(n^2/4))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 2^((n^2 + n)/(2 d)), EvenQ[d], 2^(n^2/(2 d))]]])
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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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