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A239257 Number of canyon polycubes of a given volume. 0
1, 3, 7, 16, 35, 73, 151, 304, 604, 1198, 2362, 4637, 9117, 17954, 35350, 69760, 137959, 273213, 542015, 1076870, 2141996, 4265350, 8501015, 16954408, 33833943, 67549763, 134912857, 269532456, 538603324, 1076479708, 2151817116, 4301833827, 8600826484 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A polycube P is a canyon polycube if the following conditions are satisfied: - if the cell with coordinates (a,b,c) belongs to P, then the cell with coordinate (a-1,b,c) also belongs to P (for a>1); - for each cell with coordinates (a,b,c) in P such that a = max { a' , (a',b,c) in P }, either a = max { a' , (a',b',c) in P } or a = max { a' , (a',b,c') in P }.

LINKS

Table of n, a(n) for n=1..33.

Christophe Carré et al., Dirichlet convolution and enumeration of pyramid polycubes, arXiv:1311.4836 [math.CO], 2013.

C. Carre, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015) 15.11.4

FORMULA

If n(i,j,h,v) denotes the number of canyons of height h, volume v such that the highest plateau has volume i * j, the following recurrence relation holds: n(i,j,h,v) = sum_{0 <= a <= i} sum_{0 <= b <= j} binomial(i+a,i) binomial(j+h,j) n(i+a,j+b,h-1,v-i*j).

MAPLE

calc2can:=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+binomial(i+a, a)*binomial(j+b, b)*calc2can(i+a, j+b, k-1, l-i*j);

                b:=b+1;

            od;

            a:=a+1;

        od;

        s;

    fi;

end;

comptec:=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*calc2can(i, j, k, l);

                j:=j+1;

            od:

            i:=i+1;

        od;

    od;

    s;

end;

enumc:=[seq(comptec(ii), ii=1..485)]:

convert([seq(enumc[i]*x^i, i=1..nops(%))], `+`):seriec:=subs(t=1, %);

MATHEMATICA

calc2can[i_, j_, k_, l_] := calc2can[i, j, k, l] = Module[{}, Which[l < 0, 0, i*j*k > l, 0, k == 1, If [i*j == l, 1, 0], True, s = 0; a = 0; b = 0;

While[(i + a)*j*(k - 1) <= l - i*j, b = 0; While[(i + a)*(j + b)*(k - 1) <= l - i*j, s = s + Binomial[i + a, a]*Binomial[j + b, b]*calc2can[i + a, j + b, k - 1, l - i*j]; b++]; a++]; s]];

comptec[l_] := Module[{s = 0}, For[k = 1, k <= l, k++, i = 1; While[i*k <= l, j = 1; While[i*k*j <= l, s = s + t^k*calc2can[i, j, k, l]; j++]; i++] ]; s ];

Array[comptec, 40] /. t -> 1 (* Jean-François Alcover, Dec 05 2017, translated from Maple *)

CROSSREFS

Cf. A229915, A227926.

Sequence in context: A026734 A026767 A240740 * A268394 A238913 A133124

Adjacent sequences:  A239254 A239255 A239256 * A239258 A239259 A239260

KEYWORD

nonn

AUTHOR

Matthieu Deneufchâtel, Mar 13 2014

STATUS

approved

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Last modified May 24 04:25 EDT 2019. Contains 323528 sequences. (Running on oeis4.)