A060734 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.

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

Offset: 1

Author

Frank Ellermann, Apr 23 2001

Keywords

• nonn
• tabl

Comments

A simple permutation of natural numbers.

Parity of the sequence is given by A057211 (n-th run has length n). - Jeremy Gardiner, Dec 26 2008

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

Examples

			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
		

Links

• Alois P. Heinz, Rows n = 1..141 of triangle, flattened
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
• Eric W. Weisstein, MathWorld: Pairing functions
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A060736. Inverse: A064790.

Cf. A185725, A185726, A185728.

Programs

Maple

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);

Mathematica

f[n_, k_]:=k^2-n+1/; k>=n;
f[n_, k_]:=(n-1)^2+k/; kClark Kimberling, Feb 01 2011 *)

Formula

T(n,k) = (n-1)^2+k, T(k, n)=n^2+1-k, 1 <= k <= n.

From Clark Kimberling, Feb 01 2011: (Start)

T(1,k) = k^2 (A000290).

T(n,n) = n^2-n+1 (A002061).

T(n,1) = (n-1)^2+1 (A002522). (End)

Extensions

Corrected by Jeremy Gardiner, Dec 26 2008