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A060734
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Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are read by antidiagonals downwards.
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15
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1, 4, 2, 9, 3, 5, 16, 8, 6, 10, 25, 15, 7, 11, 17, 36, 24, 14, 12, 18, 26, 49, 35, 23, 13, 19, 27, 37, 64, 48, 34, 22, 20, 28, 38, 50, 81, 63, 47, 33, 21, 29, 39, 51, 65, 100, 80, 62, 46, 32, 30, 40, 52, 66, 82, 121, 99, 79, 61, 45, 31, 41, 53, 67, 83, 101
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OFFSET
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1,2
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COMMENTS
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A simple permutation of natural numbers.
The square with corners T(1,1)=1 and T(n,n)=n^2-n+1 is occupied by the numbers 1,2,...,n^2. - Clark Kimberling, Feb 01 2011
a(n) is pairing function - function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} - the set of integer positive numbers. - Boris Putievskiy, Dec 17 2012
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LINKS
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FORMULA
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T(n,k) = (n-1)^2+k, T(k, n)=n^2+1-k, 1 <= k <= n.
T(n,1) = (n-1)^2+1 (A002522). (End)
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EXAMPLE
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Northwest corner:
.1 4 9 16 .. => a(1) = 1
.2 3 8 15 .. => a(2) = 4, a(3) = 2
.5 6 7 14 .. => a(4) = 9, a(5) = 3, a(6) = 5
10 11 12 13 .. => a(7) = 16, a(8) = 8, a(9) = 6, a(10)=10
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MAPLE
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T:= (n, k)-> `if`(n<=k, k^2-n+1, (n-1)^2+k):
seq(seq(T(n, d-n), n=1..d-1), d=2..15);
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MATHEMATICA
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f[n_, k_]:=k^2-n+1/; k>=n;
f[n_, k_]:=(n-1)^2+k/; k<n;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten (* Clark Kimberling, Feb 01 2011 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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