login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=0..91.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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