cp's OEIS Frontend

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.

%I A152124 #13 Jul 10 2015 19:38:02
%S A152124 1,2,8,36,162,746,3420,15738,72352,332850,1530928,7042422,32394478,
%T A152124 149015678,685471704,3153185542,14504703924,66721946584,306922286796,
%U A152124 1411848979422,6494534685710,29874996141112,137425609255358,632160693109496,2907952479953454
%N 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.
%H A152124 Hugo van der Sanden, <a href="/A152124/b152124.txt">Table of n, a(n) for n = 0..100</a>
%F A152124 Let u(n) represent the number of decompositions of a 1 x n rectangle.
%F A152124 Then: u(n) = 2^(n-1) for n > 0, u(n) = 1 for n = 0.
%F A152124 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.
%F A152124 Then: t(n) = t(n-2) + sum_1^{(n-3)/2}{ 2 u(i)^2 t(n-2i-2) }
%F A152124 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.
%F A152124 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.)
%F A152124 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.
%F A152124 This gives a much faster way to calculate values for the sequence (as s(0, n)).
%e A152124 Example: the partitions comprising a(2)=8 are:
%e A152124 AA AA AB AA AB BC BA AB
%e A152124 AA BB AB BC AC AA CA CD
%e A152124 I.e., exactly those of A078469(2)=12 except for the 4 rotations of the one partition that includes an asymmetric piece:
%e A152124 AA
%e A152124 AB
%Y A152124 Cf. A078057, A152113.
%K A152124 nonn
%O A152124 0,2
%A A152124 _Hugo van der Sanden_, Mar 23 2009
%E A152124 Entries changed by _N. J. A. Sloane_ to match the b-file, Oct 04 2010