A152124 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.
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
Offset: 0
Keywords
Examples
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
Links
- Hugo van der Sanden, Table of n, a(n) for n = 0..100
Formula
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)).
Extensions
Entries changed by N. J. A. Sloane to match the b-file, Oct 04 2010