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A187276 Number of d+/d- diagonally convex polyominoes with n cells. 2


%S 1,2,6,19,61,196,630,2024,6499,20860,66941,214797,689201,2211347,

%T 7095226,22765414,73044113,234366327,751978494,2412768983,7741517800,

%U 24839137696,79697907919,255715662623

%N Number of d+/d- diagonally convex polyominoes with n cells.

%C A polyomino is d+ [d-] convex if the intersection of its interior with any line of slope 1 [-1] through the centers of the cells is connected.

%D M. Bousquet-Mélou and R. Brak, "Exactly Solved Models", in A. J. Guttmann, ed., Polygons, Polyominoes and Polycubes, Springer, 2009, pp. 46 & 76.

%e A(5) = 61 = A001168(5) - 2, omitting two of the orientations of the V pentomino.

%t ab[n_,m_,q_]:=Sum[q[n-m-r,k],{r,1,m},{k,m+1-r,n-m-r}]

%t bb[n_,m_,q_]:=Sum[q[n-m-r,m-r],{r,1,m-1}]+Sum[q[n-m-r,k],{r,1,m-1},{k,m-r,n-m-r}]

%t cb[n_,m_,q_]:=Sum[q[n-m-r,m-1-r],{r,1,m-2}]

%t a[n_,m_]:=0/;n<=1||m<=0

%t a[n_,m_]:=a[n,m]=Sum[(k-m)p[n-m,k],{k,m+1,n-m}]+ab[n,m,b]+2ab[n,m,c]+Sum[(r-1)c[n-m-r,m+1-r],{r,2,m}]

%t b[1,1]=1;

%t b[n_,m_]:=0/;n<=1||m<=0

%t b[n_,m_]:=b[n,m]=2Sum[p[n-m,k],{k,m,n-m}]+bb[n,m,b]+2bb[n,m,c]+2Sum[(r-1)c[n-m-r,m-r],{r,2,m-1}]

%t c[n_,m_]:=0/;n<=1||m<=0

%t c[n_,m_]:=c[n,m]=p[n-m,m-1]+cb[n,m,b]+2cb[n,m,c]+Sum[(r-1)c[n-m-r,m-1-r],{r,2,m-2}]

%t p[n_,m_]:=a[n,m]+b[n,m]+c[n,m]

%t Table[Sum[p[n,m],{m,(n+1)/2}],{n,20}]

%Y Cf. A001168 (fixed polyominoes), A001169 (row-convex polyominoes).

%K nonn

%O 1,2

%A _David Bevan_, Mar 07 2011

%E Typo in example corrected by _David Bevan_, Mar 23 2013

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Last modified October 21 20:44 EDT 2019. Contains 328315 sequences. (Running on oeis4.)