login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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)).

EXAMPLE

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

CROSSREFS

Cf. A078057, A152113.

Sequence in context: A228197 A326244 A027743 * A147722 A089387 A206902

Adjacent sequences:  A152121 A152122 A152123 * A152125 A152126 A152127

KEYWORD

nonn

AUTHOR

Hugo van der Sanden, Mar 23 2009

EXTENSIONS

Entries changed by N. J. A. Sloane to match the b-file, Oct 04 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 16 01:43 EST 2019. Contains 330013 sequences. (Running on oeis4.)