cp's OEIS Frontend

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.

%I A358289 #33 Nov 29 2022 01:34:07
%S A358289 0,4,16,140,2804,161036,27803749,14314228378,21838347160809,
%T A358289 99704315229167288,1367135978051264146578,56578717186086829451888706,
%U A358289 7065692298178203128922479762418,2670113158846160742372913777087464324,3052313665715695874527667027409186333152556
%N 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.
%H A358289 Anthony J. Guttmann and Iwan Jensen, <a href="/A358289/b358289.txt">Table of n, a(n) for n = 1..17</a>
%H A358289 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 A358289 a(2*n)/2 = A348456(n).
%e A358289 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 A358289 For n = 3 we must choose a connected region of area 4 with a connected complement of area 5.
%e A358289 The possibilities are
%e A358289   XXO      XXX      XXX
%e A358289   XXO      XOO      OXO
%e A358289   OOO      OOO      OOO
%e A358289   4 ways   8 ways   4 ways
%e A358289 so a(3) = 16.
%Y A358289 Cf. A348456.
%K A358289 nonn
%O A358289 1,2
%A A358289 _N. J. A. Sloane_, Nov 25 2022