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A230119 Numbers of quasi-pyramid polycubes of a given volume (number of atomic cells) 0
3, 9, 23, 47, 91, 169, 291, 494, 815, 1295, 2043, 3155, 4775, 7165, 10599, 15458, 22455, 32145, 45659, 64519, 90247, 125493, 173515, 238153, 325423, 442169, 597575, 804203, 1077283, 1436593, 1908571, 2525611, 3329391, 4373969, 5726611, 7472763, 9721983, 12608475, 16305179, 21027921, 27043631, 34689187, 44385995, 56652421, 72137483, 91645813, 116167379, 146932651, 185456419, 233594265 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

A quasi-pyramid polycube is an object obtained from a pyramid by removing all the cells with coordinates (a,b,c) with 1 <= a <= h for a fixed pair (b,c) chosen among the triples (h,b,c) such that there is a cell with coordinates (h,b,c) in the pyramid ((h,b,c) belongs to the highest plateau of the pyramid).

If Q(x) denotes the generating function of the quasi-pyramids and P(x,h) the generating function of the pyramids counted by height, then the x^(-h) P(x,h) converges when h goes to infinity and the limit is Q(x) + x/(1-x).

LINKS

Table of n, a(n) for n=1..50.

MAPLE

calcPyr:=proc(i, j, k, l) option remember;

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+(a+1)*(b+1)*calcPyr(i+a, j+b, k-1, l-i*j);

b:=b+1;

od;

a:=a+1;

od;

s;

fi;

end;

countPyr:=proc(l)

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*calcRecPyr(i, j, k, l);

j:=j+1;

od:

i:=i+1;

od;

od;

s;

end;

enum=[seq(countPyr(ii), ii=1..200)];

serie_quasiPyr:=proc(l) global enum; local ii;

map(coeff, enump, t^l);

select(x->x>0, %);

sum(t^(ii-1)*%[ii], ii=1..nops(%));

end;

serie_quasiPyr(100):

[1, seq(coeff(%, t^ii)-1, ii=1..50)];

CROSSREFS

A229914, A227926

Sequence in context: A210027 A018555 A147271 * A146693 A227259 A064551

Adjacent sequences: A230116 A230117 A230118 * A230120 A230121 A230122

KEYWORD

nonn

AUTHOR

Matthieu Deneufchâtel, Oct 10 2013

STATUS

approved

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Last modified January 30 02:06 EST 2023. Contains 359939 sequences. (Running on oeis4.)