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A152124
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Number of partitions of a 2 x n rectangle into connected pieces consisting of unit squares cut along lattice lines (like a 2-d analog of a partition into integers) in which each piece has rotational symmetry.
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3
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1, 2, 8, 36, 162, 746, 3420, 15738, 72352, 332850, 1530928, 7042422, 32394478, 149015678, 685471704, 3153185542, 14504703924, 66721946584, 306922286796, 1411848979422, 6494534685710, 29874996141112, 137425609255358, 632160693109496, 2907952479953454
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OFFSET
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0,2
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LINKS
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FORMULA
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Let u(n) represent the number of decompositions of a 1 x n rectangle.
Then: u(n) = 2^(n-1) for n > 0, u(n) = 1 for n = 0.
Let t(n) represent the number of decompositions of a 2 x n rectangle such that a single piece includes the whole of the leftmost and rightmost columns.
Then: t(n) = t(n-2) + sum_1^{(n-3)/2}{ 2 u(i)^2 t(n-2i-2) }
Let s(m, n) represent the number of decompositions of a 2 x n rectangle with a 1 x m spike attached to the side.
Then for m > 0: s(m, n) = sum_1^m{ s(m-i, n) } + sum_1^n{ s(i, n-i) } + sum_m^{(n+m-1)/2}{ u(i-m) sum_1^{n+m-2i}{ t(j) s(i, n+m-2i-j) } } and for m = 0: s(m, n) = sum_1^n{ s(i, n-i) } + sum_1^n{ t(i) s(0, n-i) } + sum_1^{(n-1)/2){ u(i) sum_1^{n-2i}{ t(j) s(i, n-2i-j) } } (Note that these sums can be taken to infinity if the functions are defined as zero when any argument is negative.)
We get t(n) = [ 0 1 1 1 1 3 3 13 13 59 59 269 269 1227 1227 5597 5597 25531 ... ] = A052984((n - 3) / 2) with recurrence a(n) = 5a(n-1)-2a(n-2), a(0) = 1, a(1) = 3.
This gives a much faster way to calculate values for the sequence (as s(0, n)).
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EXAMPLE
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Example: the partitions comprising a(2)=8 are:
AA AA AB AA AB BC BA AB
AA BB AB BC AC AA CA CD
I.e., exactly those of A078469(2)=12 except for the 4 rotations of the one partition that includes an asymmetric piece:
AA
AB
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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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