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

%I A187276 #11 Sep 01 2013 16:24:54
%S A187276 1,2,6,19,61,196,630,2024,6499,20860,66941,214797,689201,2211347,
%T A187276 7095226,22765414,73044113,234366327,751978494,2412768983,7741517800,
%U A187276 24839137696,79697907919,255715662623
%N A187276 Number of d+/d- diagonally convex polyominoes with n cells.
%C A187276 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 A187276 M. Bousquet-Mélou and R. Brak, "Exactly Solved Models", in A. J. Guttmann, ed., Polygons, Polyominoes and Polycubes, Springer, 2009, pp. 46 & 76.
%e A187276 A(5) = 61 = A001168(5) - 2, omitting two of the orientations of the V pentomino.
%t A187276 ab[n_,m_,q_]:=Sum[q[n-m-r,k],{r,1,m},{k,m+1-r,n-m-r}]
%t A187276 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 A187276 cb[n_,m_,q_]:=Sum[q[n-m-r,m-1-r],{r,1,m-2}]
%t A187276 a[n_,m_]:=0/;n<=1||m<=0
%t A187276 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 A187276 b[1,1]=1;
%t A187276 b[n_,m_]:=0/;n<=1||m<=0
%t A187276 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 A187276 c[n_,m_]:=0/;n<=1||m<=0
%t A187276 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 A187276 p[n_,m_]:=a[n,m]+b[n,m]+c[n,m]
%t A187276 Table[Sum[p[n,m],{m,(n+1)/2}],{n,20}]
%Y A187276 Cf. A001168 (fixed polyominoes), A001169 (row-convex polyominoes).
%K A187276 nonn
%O A187276 1,2
%A A187276 _David Bevan_, Mar 07 2011
%E A187276 Typo in example corrected by _David Bevan_, Mar 23 2013