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 A227925 Triangle read by rows: number of espalier polycubes counted by height and volume 4
 1, 1, 2, 1, 2, 2, 1, 3, 4, 2, 1, 2, 5, 4, 2, 1, 4, 8, 7, 4, 2, 1, 2, 8, 10, 7, 4, 2, 1, 4, 13, 14, 12, 7, 4, 2, 1, 3, 12, 19, 16, 12, 7, 4, 2, 1, 4, 17, 26, 25, 18, 12, 7, 4, 2, 1, 2, 16, 29, 32, 27, 18, 12, 7, 4, 2, 1, 6, 24, 41, 45, 38, 29, 18, 12, 7, 4, 2, 1, 2, 19, 44, 55, 51, 40, 29, 18, 12, 7, 4, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A pyramid polycube is obtained by gluing together horizontal plateaux (parallelepipeds of height 1) in such a way that (0,0,0) belongs to the first plateau and each cell with coordinate (0,b,c) belonging to the first plateau is such that b , c >= 0. If the cell with coordinates (a,b,c) belongs to the (a+1)-st plateau (a>0), then the cell with coordinates (a-1, b, c) belongs to the a-th plateau. An espalier polycube is a special pyramid such that each plateau contains the cell with coordinate (a,0,0). LINKS FORMULA The number n_{i,j,h,v} of espaliers of volume v, height h and such that the highest plateau has volume i * j is given by the recurrence: n_{i,j,h,v} = \sum_{0 <= a <= (i*j*h-v)/((h-1)j)} \sum_{0 <= b <= (j(h(i+a)-a)-v)/((i+a)(k-1))} n_{i+a,j+a,h-1,v-ij} The number of espaliers of volume v and height h is given by n_{h,v}=\sum_{i*j<=v}n_{i,j,h,v} MAPLE calcRecEsp:=proc(i, j, k, l) option remember; ## Compute the number n_{i, j, k, l} if (l<0) then 0 elif (i*j*k>l) then 0 elif k=1 then if (i*j=l) then 1 else 0; fi; else s:=0; a:=0; b:=0; while ((i+a)*j*(k-1)<=l-i*j) do b:=0; while ((i+a)*(j+b)*(k-1)<=l-i*j) do s:=s+calcRecEsp(i+a, j+b, k-1, l-i*j); b:=b+1; od; a:=a+1; od; s; fi; end; compteEsp:=proc(l) ### compute \sum_{v}n_{h, v}t^v s:=0; for k to l do i:=1: while (i*k<=l) do j:=1; while (i*k*j<=l) do s:=s+t^k*calcRecEsp(i, j, k, l); j:=j+1; od: i:=i+1; od; od; s; end; [1, seq(op(convert(compteEsp(ii), list)), ii=2..200)]; CROSSREFS The numbers of espaliers counted by volume are given by A229915 Sequence in context: A029287 A055184 A238190 * A035388 A255716 A177954 Adjacent sequences:  A227922 A227923 A227924 * A227926 A227927 A227928 KEYWORD nonn,tabl AUTHOR Matthieu Deneufchâtel, Oct 09 2013 STATUS approved

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Last modified November 20 02:57 EST 2018. Contains 317371 sequences. (Running on oeis4.)