%I #32 Feb 28 2020 05:33:14

%S 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,

%T 27,37,64,48,34,22,20,28,38,50,81,63,47,33,21,29,39,51,65,100,80,62,

%U 46,32,30,40,52,66,82,121,99,79,61,45,31,41,53,67,83,101

%N 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.

%C A simple permutation of natural numbers.

%C Parity of the sequence is given by A057211 (n-th run has length n). - _Jeremy Gardiner_, Dec 26 2008

%C 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

%C 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

%H Alois P. Heinz, <a href="/A060734/b060734.txt">Rows n = 1..141 of triangle, flattened</a>

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PairingFunction.html">MathWorld: Pairing functions</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

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

%F From _Clark Kimberling_, Feb 01 2011: (Start)

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

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

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

%e Northwest corner:

%e .1 4 9 16 .. => a(1) = 1

%e .2 3 8 15 .. => a(2) = 4, a(3) = 2

%e .5 6 7 14 .. => a(4) = 9, a(5) = 3, a(6) = 5

%e 10 11 12 13 .. => a(7) = 16, a(8) = 8, a(9) = 6, a(10)=10

%p T:= (n,k)-> `if`(n<=k, k^2-n+1, (n-1)^2+k):

%p seq(seq(T(n, d-n), n=1..d-1), d=2..15);

%t f[n_, k_]:=k^2-n+1/; k>=n;

%t f[n_, k_]:=(n-1)^2+k/; k<n;

%t TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]

%t Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten (* _Clark Kimberling_, Feb 01 2011 *)

%Y Cf. A060736. Inverse: A064790.

%Y Cf. A185725, A185726, A185728.

%K nonn,tabl

%O 1,2

%A _Frank Ellermann_, Apr 23 2001

%E Corrected by _Jeremy Gardiner_, Dec 26 2008