A355880 -id:A355880 - cp's OEIS frontend

A355966 Number of 3-dimensional polyominoes (or polycubes) with n cells that have cavities (inclusions of empty space).

20, 404, 6164, 75917, 835491

Offset: 11

Gleb Ivanov, Jul 21 2022

Even if two polycubes are mirror images of each other, they are considered different for this sequence.
Polycubes with less than 11 cells can't have cavities.
Largest enclosed volume >= A355880(n-5) for polycubes with n cells.

Gleb Ivanov, Python program for n = 11..14
Wikipedia, Polycube

Cf. A001419, A000162, A038119, A355880.

Cf. A357083 (without distinguished reflections).

A377133 Triangle read by rows: T(n,k) is the maximum volume of an integer-sided box that can be made from a piece of paper of size n X k by cutting away identical squares at each corner and folding up the sides, n >= 3, 3 <= k <= n.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 4, 8, 12, 16, 5, 10, 15, 20, 25, 6, 12, 18, 24, 30, 36, 7, 14, 21, 28, 35, 42, 50, 8, 16, 24, 32, 40, 48, 60, 72, 9, 18, 27, 36, 45, 56, 70, 84, 98, 10, 20, 30, 40, 50, 64, 80, 96, 112, 128, 11, 22, 33, 44, 55, 72, 90, 108, 126, 144, 162, 12, 24

Offset: 3

Felix Huber, Oct 25 2024

For a sketch see linked illustration "Box made from nXk-paper".

The first few rows follow (n-2) * (k-2), so the initial terms are the same as in A075362. The first difference is at T(9,9) = 50 which is greater than 7 * 7.

			Triangle T(n,k) begins:
   n\k   3     4     5     6     7     8     9    10    11    12    13 ...
   3     1
   4     2     4
   5     3     6     9
   6     4     8    12    16
   7     5    10    15    20    25
   8     6    12    18    24    30    36
   9     7    14    21    28    35    42    50
  10     8    16    24    32    40    48    60    72
  11     9    18    27    36    45    56    70    84    98
  12    10    20    30    40    50    64    80    96   112   128
  13    11    22    33    44    55    72    90   108   126   144   162

Felix Huber, Rows n = 3..142 of triangle, flattened
Felix Huber, Box made from nXk-paper

Cf. A075362, A355880, A375580, A375785, A375785.

A377113:=proc(n,k)
   local a,x,V;
   a:=0;
   for x to (k-1)/2 do
      V:=x*(n-2*x)*(k-2*x);
      if V>a then
         a:=V
      fi
   od;
   return a
end proc;
seq(seq(A377113(n,k),k=3..n),n=3..14);

T(n,k) = (n-2*x)*(k-2*x)*x with x = round((n+k-(sqrt(n^2+k^2-n*k)))/6).

cp's OEIS Frontend

A355966 Number of 3-dimensional polyominoes (or polycubes) with n cells that have cavities (inclusions of empty space).

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A377133 Triangle read by rows: T(n,k) is the maximum volume of an integer-sided box that can be made from a piece of paper of size n X k by cutting away identical squares at each corner and folding up the sides, n >= 3, 3 <= k <= n.

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