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 A358289 Generalized Gerrymander sequence: number of ordered ways to divide an n X n square into two connected regions, both of area n^2/2 if n is even, or of areas (n^2-1)/2 and (n^2+1)/2 if n is odd. 3

%I #33 Nov 29 2022 01:34:07

%S 0,4,16,140,2804,161036,27803749,14314228378,21838347160809,

%T 99704315229167288,1367135978051264146578,56578717186086829451888706,

%U 7065692298178203128922479762418,2670113158846160742372913777087464324,3052313665715695874527667027409186333152556

%N Generalized Gerrymander sequence: number of ordered ways to divide an n X n square into two connected regions, both of area n^2/2 if n is even, or of areas (n^2-1)/2 and (n^2+1)/2 if n is odd.

%H Anthony J. Guttmann and Iwan Jensen, <a href="/A358289/b358289.txt">Table of n, a(n) for n = 1..17</a>

%H Anthony J. Guttmann and Iwan Jensen, <a href="https://arxiv.org/abs/2211.14482">The gerrymander sequence, or A348456</a>, arXiv:2211.14482 [math.CO], 2022.

%F a(2*n)/2 = A348456(n).

%e For n = 2, the square can be split vertically or horizontally, and then there are two ways to order the regions, so a(2) = 2*2 = 4.

%e For n = 3 we must choose a connected region of area 4 with a connected complement of area 5.

%e The possibilities are

%e XXO XXX XXX

%e XXO XOO OXO

%e OOO OOO OOO

%e 4 ways 8 ways 4 ways

%e so a(3) = 16.

%Y Cf. A348456.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Nov 25 2022

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Last modified September 28 14:22 EDT 2023. Contains 365735 sequences. (Running on oeis4.)