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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.

%I A377133 #17 Nov 17 2024 07:41:16
%S A377133 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,
%T A377133 42,50,8,16,24,32,40,48,60,72,9,18,27,36,45,56,70,84,98,10,20,30,40,
%U A377133 50,64,80,96,112,128,11,22,33,44,55,72,90,108,126,144,162,12,24
%N 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.
%C A377133 For a sketch see linked illustration "Box made from nXk-paper".
%C A377133 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.
%H A377133 Felix Huber, <a href="/A377133/b377133.txt">Rows n = 3..142 of triangle, flattened</a>
%H A377133 Felix Huber, <a href="/A377133/a377133.pdf">Box made from nXk-paper</a>
%F A377133 T(n,k) = (n-2*x)*(k-2*x)*x with x = round((n+k-(sqrt(n^2+k^2-n*k)))/6).
%e A377133 Triangle T(n,k) begins:
%e A377133    n\k   3     4     5     6     7     8     9    10    11    12    13 ...
%e A377133    3     1
%e A377133    4     2     4
%e A377133    5     3     6     9
%e A377133    6     4     8    12    16
%e A377133    7     5    10    15    20    25
%e A377133    8     6    12    18    24    30    36
%e A377133    9     7    14    21    28    35    42    50
%e A377133   10     8    16    24    32    40    48    60    72
%e A377133   11     9    18    27    36    45    56    70    84    98
%e A377133   12    10    20    30    40    50    64    80    96   112   128
%e A377133   13    11    22    33    44    55    72    90   108   126   144   162
%p A377133 A377113:=proc(n,k)
%p A377133    local a,x,V;
%p A377133    a:=0;
%p A377133    for x to (k-1)/2 do
%p A377133       V:=x*(n-2*x)*(k-2*x);
%p A377133       if V>a then
%p A377133          a:=V
%p A377133       fi
%p A377133    od;
%p A377133    return a
%p A377133 end proc;
%p A377133 seq(seq(A377113(n,k),k=3..n),n=3..14);
%Y A377133 Cf. A075362, A355880, A375580, A375785, A375785.
%K A377133 nonn,tabl,easy
%O A377133 3,2
%A A377133 _Felix Huber_, Oct 25 2024