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.

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

Author

Felix Huber, Oct 25 2024

Keywords

• nonn
• tabl
• easy

Comments

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.

Examples

			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
		

Links

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

Crossrefs

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

Programs

Maple

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);

Formula

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