cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Matthieu Deneufchâtel

Matthieu Deneufchâtel's wiki page.

Matthieu Deneufchâtel has authored 11 sequences. Here are the ten most recent ones:

A239257 Number of canyon polycubes of a given volume.

Original entry on oeis.org

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
Offset: 1

Views

Author

Matthieu Deneufchâtel, Mar 13 2014

Keywords

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

Crossrefs

Cf. A229915, A227926.

Programs

  • 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 *)

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).

A229925 Numbers of espalier polycubes of a given volume in dimension 5.

Original entry on oeis.org

1, 5, 9, 23, 31, 71, 87, 173, 223, 379, 471, 801, 951, 1495, 1851, 2736, 3282, 4832, 5708, 8126, 9704, 13290, 15694, 21496, 25038, 33396, 39330, 51452
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 03 2013

Keywords

Comments

A (d+1)-pyramid polycube is a (d+1)-polycube obtained by gluing together horizontal (d+1)-plateaux (parallelepipeds of height 1) in such a way that the cell (0,0,...,0) belongs to the first plateau and each cell with coordinates (0,n_1,...,n_d) belonging to the first plateau is such that n_1 , ... , n_d >= 0.
If the cell with coordinates (n_0,n_1,...,n_d) belongs to the (n_0+1)-st plateau (n_0>0), then the cell with coordinates (n_0-1, n_1, ... ,n_d) belongs to the n_0-th plateau.
A (d+1)-espalier is a (d+1)-pyramid such that each plateau contains the cell (n_0,0,...,0).

Crossrefs

Cf. A227925, A229915.

A229924 Numbers of pyramid polycubes of a given volume in dimension 5.

Original entry on oeis.org

1, 5, 13, 35, 81, 165, 329, 587, 1059, 1807, 2939, 4633, 7431, 11391, 17091, 25372, 37343, 53913, 77393, 109273, 153857, 214409, 295293, 403545, 551715, 745643, 1001721, 1339677, 1784109
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 03 2013

Keywords

Comments

A (d+1)-pyramid polycube is a (d+1)-polycube obtained by gluing together horizontal (d+1)-plateaux (parallelepipeds of height 1) in such a way that the cell (0,0,...,0) belongs to the first plateau and each cell with coordinates (0,n_1,...,n_d) belonging to the first plateau is such that n_1 , ... , n_d >= 0.
If the cell with coordinates (n_0,n_1,...,n_d) belongs to the (n_0+1)-st plateau (n_0>0), then the cell with coordinates (n_0-1, n_1, ... ,n_d) belongs to the n_0-th plateau.

Crossrefs

Cf. A229914, A227926.

A229917 Numbers of espalier polycubes of a given volume in dimension 4.

Original entry on oeis.org

1, 4, 7, 16, 22, 46, 58, 107, 140, 227, 287, 464, 563, 851, 1067, 1530, 1866, 2661, 3198, 4428, 5361, 7185, 8613, 11524, 13639, 17839, 21272, 27359, 32300, 41369, 48512, 61311, 72105, 89904, 105226, 130834, 152164, 187297, 218356, 266444, 309125, 375995, 434670, 525045, 607329, 728256, 839874, 1004938
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 03 2013

Keywords

Comments

A (d+1)-pyramid polycube is a (d+1)-polycube obtained by gluing together horizontal (d+1)-plateaux (parallelepipeds of height 1) in such a way that the cell (0,0,...,0) belongs to the first plateau and each cell with coordinates (0,n_1,...,n_d) belonging to the first plateau is such that n_1 , ... , n_d >= 0.
If the cell with coordinates (n_0,n_1,...,n_d) belongs to the (n_0+1)-st plateau (n_0>0), then the cell with coordinates (n_0-1, n_1, ... ,n_d) belongs to the n_0-th plateau.
A (d+1)-espalier is a (d+1)-pyramid such that each plateau contains the cell (n_0,0,...,0).

Crossrefs

Cf. A229915, A227925.

A229916 Numbers of pyramid polycubes of a given volume in dimension 4.

Original entry on oeis.org

1, 4, 10, 25, 55, 109, 211, 371, 651, 1092, 1767, 2775, 4383, 6666, 9990, 14806, 21695, 31265, 44852, 63365, 89132, 124250, 171560, 235140, 321661, 435670, 587219, 787663, 1051669, 1396309, 1848190, 2432288, 3192615, 4174278, 5435945, 7054030, 9134731, 11779666, 15152584, 19436993, 24867486
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 03 2013

Keywords

Comments

A (d+1)-pyramid polycube is a (d+1)-polycube obtained by gluing together horizontal (d+1)-plateaux (parallelepipeds of height 1) in such a way that the cell with coordinates (0,0,...,0) belongs to the first plateau and each cell with coordinates (0,n_1,...,n_d) belonging to the first plateau is such that n_1 , ... , n_d >= 0.
If the cell with coordinates (n_0,n_1,...,n_d) belongs to the (n_0+1)-st plateau (n_0>0), then the cell with coordinates (n_0-1, n_1, ... ,n_d) belongs to the n_0-th plateau.

Crossrefs

Cf. A229914, A227926.

A230119 Numbers of quasi-pyramid polycubes of a given volume (number of atomic cells).

Original entry on oeis.org

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
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 10 2013

Keywords

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).

Crossrefs

A229914, A227926

Programs

  • 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)];

A230118 Numbers of quasi-espalier polycubes of a given volume (number of atomic cells).

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 29, 42, 61, 87, 122, 167, 229, 306, 409, 538, 705, 915, 1182, 1509, 1927, 2438, 3075, 3854, 4814, 5985, 7416, 9144, 11253, 13784, 16845, 20512, 24922, 30179, 36470, 43939, 52841, 63378, 75864, 90605, 108022, 128496, 152603, 180865, 214044, 252826, 298192, 351108, 412832, 484632, 568157
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 10 2013

Keywords

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).
Quasi espaliers are espaliers from which all the cells with coordinates (a,0,0) have been removed.
If E(x,h) denotes the generating function of espalier polycubes, x^(-h) E(x,h) converges when h tends to infinity towards a series which is the generating function of quasi-espalier polycubes.

Crossrefs

Cf. A227925, A229915.

Programs

  • 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;
enum = [seq(op(convert(compteEsp(ii), list)), ii=2..200)];
serie_quasi_Esp:=proc(l)global enum;local ii;
map(coeff,enum,t^l);
select(x->x>0,%);
sum(t^(ii-1)*%[ii],ii=1..nops(%));
end;
serie_quasi_Esp(100):
[1, seq(coeff(%, t^ii)-1, ii=1..50)];

A227926 Triangle read by rows: number of pyramid polycubes counted by height and volume.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 3, 8, 4, 1, 2, 16, 10, 4, 1, 4, 22, 22, 10, 4, 1, 2, 36, 40, 24, 10, 4, 1, 4, 47, 66, 46, 24, 10, 4, 1, 3, 60, 110, 84, 48, 24, 10, 4, 1, 4, 83, 158, 144, 90, 48, 24, 10, 4, 1, 2, 100, 233, 232, 162, 92, 48, 24, 10, 4, 1, 6, 116, 327, 357, 266, 168, 92, 48, 24, 10, 4, 1, 2, 148, 444, 544, 435, 284, 170, 92, 48, 24, 10, 4, 1
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 09 2013

Keywords

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.

Links

Crossrefs

The numbers of pyramids counted by volume are given by A229914.

Programs

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

Formula

The number n_{i,j,h,v} of pyramids 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_{a=0..(i*j*h-v)/((h-1)*j)} Sum_{b=0..(j*(h*(i+a)-a)-v)/((i+a)*(k-1))} (a+1)*(b+1)*n_{i+a,j+a,h-1,v-i*j}.

A229915 Number of espalier polycubes of a given volume in dimension 3.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 34, 57, 76, 116, 150, 227, 284, 408, 520, 718, 895, 1226, 1508, 2018, 2487, 3248, 3968, 5160, 6235, 7970, 9653, 12179, 14630, 18367, 21924, 27241, 32506, 39985, 47492, 58203, 68752, 83613, 98730, 119269, 140224, 168799, 197758, 236753, 277052, 329867, 384852, 457006, 531500, 628338
Offset: 0

Views

Author

Matthieu Deneufchâtel, Oct 03 2013

Keywords

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 coordinates (0,b,c) belongs to the first plateau 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 coordinates (a,0,0).

Links

Crossrefs

Cf. A026820, A100882, A227925, A230118, A229917, A229925, A323582.

Formula

The generating function for the numbers of espaliers of height h and volumes v_1 , ... v_h is x_1^{n_1} * ... x_h^{n_h} / ((1-x_1^{n_1}) *(1-x_1^{n_1}*x_2^{n_2}) *... *(1-x_1^{n_1}*x_2^{n_2}*...x_h^{n_h})).
This sequence is obtained with x_1 = ... = x_h = p by summing over n_1>= ... >= n_h>=1 and then over h.

Extensions

a(0)=1 prepended by Seiichi Manyama, Aug 20 2020

A227925 Triangle read by rows: number of espalier polycubes counted by height and volume.

Original entry on oeis.org

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
Offset: 0

Views

Author

Matthieu Deneufchâtel, Oct 09 2013

Keywords

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).

Crossrefs

The numbers of espaliers counted by volume are given by A229915

Programs

  • 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)];

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}

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