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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.
1
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
3,2
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.
FORMULA
T(n,k) = (n-2*x)*(k-2*x)*x with x = round((n+k-(sqrt(n^2+k^2-n*k)))/6).
EXAMPLE
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
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);
CROSSREFS
KEYWORD
nonn,tabl,easy
AUTHOR
Felix Huber, Oct 25 2024
STATUS
approved