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A295229
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Number of tilings of the n X n grid, using diagonal lines to connect the grid points.
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2
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1, 6, 84, 8548, 4203520, 8590557312, 70368815480832, 2305843028004192256, 302231454912728264605696, 158456325028538104598816096256, 332306998946228986960926214931349504, 2787593149816327892769293535238052808491008
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OFFSET
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1,2
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COMMENTS
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The grids are counted up to reflection and rotation.
a(n) <= A295223(n).
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LINKS
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Peter Kagey, Table of n, a(n) for n = 1..57
Andrew Howroyd, Derivation of Formula
Peter Kagey, Example of the 6 tilings of the 2 X 2 grid.
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FORMULA
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From Andrew Howroyd, Nov 19 2017: (Start)
a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 3*2^(n^2/2) + 2*2^(n^2/4)) / 8 for n even.
a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 2^((n^2+1)/2)) / 8 for n odd. (End)
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EXAMPLE
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For n = 2, the a(2) = 6 tilings are:
//, \/, /\, \\, /\, and \/.
// // // // \/ /\
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MATHEMATICA
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Array[(2^(#^2) + 2*2^(# (# + 1)/2) + If[EvenQ@ #, 3*2^(#^2/2) + 2*2^(#^2/4), 2^((#^2 + 1)/2)])/8 &, 12] (* Michael De Vlieger, Apr 12 2018 *)
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PROG
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(PARI) a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + if(n%2, 2^((n^2+1)/2), 3*2^(n^2/2) + 2*2^(n^2/4)))/8; \\ Andrew Howroyd, Nov 19 2017
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CROSSREFS
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Cf. A054247, A295223.
Sequence in context: A293455 A334516 A331014 * A330849 A245232 A284522
Adjacent sequences: A295226 A295227 A295228 * A295230 A295231 A295232
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KEYWORD
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nonn
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AUTHOR
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Peter Kagey, Nov 18 2017
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EXTENSIONS
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a(5)-a(12) from Andrew Howroyd, Nov 19 2017
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STATUS
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approved
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