A377133 - OEIS

%I #17 Nov 17 2024 07:41:16

%S 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 42,50,8,16,24,32,40,48,60,72,9,18,27,36,45,56,70,84,98,10,20,30,40,

%U 50,64,80,96,112,128,11,22,33,44,55,72,90,108,126,144,162,12,24

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

%C 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 Felix Huber, <a href="/A377133/b377133.txt">Rows n = 3..142 of triangle, flattened</a>

%H Felix Huber, <a href="/A377133/a377133.pdf">Box made from nXk-paper</a>

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

%e Triangle T(n,k) begins:

%e    n\k   3     4     5     6     7     8     9    10    11    12    13 ...

%e    3     1

%e    4     2     4

%e    5     3     6     9

%e    6     4     8    12    16

%e    7     5    10    15    20    25

%e    8     6    12    18    24    30    36

%e    9     7    14    21    28    35    42    50

%e   10     8    16    24    32    40    48    60    72

%e   11     9    18    27    36    45    56    70    84    98

%e   12    10    20    30    40    50    64    80    96   112   128

%e   13    11    22    33    44    55    72    90   108   126   144   162

%p A377113:=proc(n,k)

%p    local a,x,V;

%p    a:=0;

%p    for x to (k-1)/2 do

%p       V:=x*(n-2*x)*(k-2*x);

%p       if V>a then

%p          a:=V

%p       fi

%p    od;

%p    return a

%p end proc;

%p seq(seq(A377113(n,k),k=3..n),n=3..14);

%Y Cf. A075362, A355880, A375580, A375785, A375785.

%K nonn,tabl,easy

%O 3,2

%A _Felix Huber_, Oct 25 2024