| 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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