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A322550
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Table read by ascending antidiagonals: T(n, k) is the minimum number of cubes necessary to fill a right square prism with base area n^2 and height k.
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2
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1, 4, 2, 9, 1, 3, 16, 18, 12, 4, 25, 4, 1, 2, 5, 36, 50, 48, 36, 20, 6, 49, 9, 75, 1, 45, 3, 7, 64, 98, 4, 100, 80, 2, 28, 8, 81, 16, 147, 18, 1, 12, 63, 4, 9, 100, 162, 192, 196, 180, 150, 112, 72, 36, 10, 121, 25, 9, 4, 245, 1, 175, 2, 3, 5, 11, 144, 242, 300, 324, 320, 294, 252, 200, 144, 90, 44, 12
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OFFSET
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1,2
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LINKS
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FORMULA
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T(n, k) = n^2*k/gcd(n, k)^3.
First column: T(n, 1) = A000290(n).
First row of the table: T(1, n) = A000027(n).
Main diagonal of the table: T(n, n) = A000012(n).
Superdiagonal of the table: T(n, n + 1) = A011379(n).
Subdiagonal of the table: T(n, n - 1) = A045991(n).
X(n, k) = T(n + 1 - k, k).
Diagonal of the triangle: X(n, n) = A000027(n).
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EXAMPLE
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The table T starts in row n = 1 with columns k >= 1 as:
1 2 3 4 5 6 7 8 9 ...
4 1 12 2 20 3 28 4 36 ...
9 18 1 36 45 2 63 72 3 ...
16 4 48 1 80 12 112 2 144 ...
25 50 75 100 1 150 175 200 225 ...
36 9 4 18 180 1 252 36 12 ...
49 98 147 196 245 294 1 392 441 ...
64 16 192 4 320 48 448 1 576 ...
81 162 9 324 405 18 567 648 1 ...
...
The triangle X(n, k) begins
n\k| 1 2 3 4 5 6 7 8 9
---+----------------------------------------------------
1 | 1
2 | 4 2
3 | 9 1 3
4 | 16 18 12 4
5 | 25 4 1 2 5
6 | 36 50 48 36 20 6
7 | 49 9 75 1 45 3 7
8 | 64 98 4 100 80 2 28 8
9 | 81 16 147 18 1 12 63 4 9
...
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MAPLE
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a := (n, k) -> (n+1-k)^2*k/gcd(n+1-k, k)^3: seq(seq(a(n, k), k = 1 .. n), n = 1 .. 12)
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MATHEMATICA
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T[n_, k_]:=n^2*k/GCD[n, k]^3; Flatten[Table[T[n-k+1, k], {n, 12}, {k, n}]]
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PROG
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(GAP) Flat(List([1..12], n->List([1..n], k->(n+1-k)^2*k/GcdInt(n+1-k, k)^3)));
(Magma) [[(n+1-k)^2*k/Gcd(n+1-k, k)^3: k in [1..n]]: n in [1..12]]; // triangle output
(Maxima) sjoin(v, j) := apply(sconcat, rest(join(makelist(j, length(v)), v)))$ display_triangle(n) := for i from 1 thru n do disp(sjoin(makelist((i+1-j)^2*j/gcd(i+1-j, j)^3, j, 1, i), " ")); display_triangle(12);
(PARI)
T(n, k) = (n+1-k)^2*k/gcd(n+1-k, k)^3;
tabl(nn) = for(i=1, nn, for(j=1, i, print1(T(i, j), ", ")); print);
tabl(12) \\ triangle output
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CROSSREFS
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Cf. A011379 (superdiagonal of the table), A045991 (subdiagonal of the table).
Cf. A320043 (row sums of the triangle).
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KEYWORD
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AUTHOR
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STATUS
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approved
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