cp's OEIS Frontend

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.

%I A060734 #33 Feb 16 2025 08:32:44
%S A060734 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 A060734 27,37,64,48,34,22,20,28,38,50,81,63,47,33,21,29,39,51,65,100,80,62,
%U A060734 46,32,30,40,52,66,82,121,99,79,61,45,31,41,53,67,83,101
%N 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.
%C A060734 A simple permutation of natural numbers.
%C A060734 Parity of the sequence is given by A057211 (n-th run has length n). - _Jeremy Gardiner_, Dec 26 2008
%C A060734 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 A060734 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 A060734 Alois P. Heinz, <a href="/A060734/b060734.txt">Rows n = 1..141 of triangle, flattened</a>
%H A060734 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H A060734 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/PairingFunction.html">MathWorld: Pairing functions</a>
%H A060734 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A060734 T(n,k) = (n-1)^2+k, T(k, n)=n^2+1-k, 1 <= k <= n.
%F A060734 From _Clark Kimberling_, Feb 01 2011: (Start)
%F A060734 T(1,k) = k^2 (A000290).
%F A060734 T(n,n) = n^2-n+1 (A002061).
%F A060734 T(n,1) = (n-1)^2+1 (A002522). (End)
%e A060734 Northwest corner:
%e A060734 .1  4  9 16 ..  => a(1) =  1
%e A060734 .2  3  8 15 ..  => a(2) =  4, a(3) = 2
%e A060734 .5  6  7 14 ..  => a(4) =  9, a(5) = 3, a(6) = 5
%e A060734 10 11 12 13 ..  => a(7) = 16, a(8) = 8, a(9) = 6, a(10)=10
%p A060734 T:= (n,k)-> `if`(n<=k, k^2-n+1, (n-1)^2+k):
%p A060734 seq(seq(T(n, d-n), n=1..d-1), d=2..15);
%t A060734 f[n_, k_]:=k^2-n+1/; k>=n;
%t A060734 f[n_, k_]:=(n-1)^2+k/; k<n;
%t A060734 TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
%t A060734 Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten (* _Clark Kimberling_, Feb 01 2011 *)
%Y A060734 Cf. A060736. Inverse: A064790.
%Y A060734 Cf. A185725, A185726, A185728.
%K A060734 nonn,tabl
%O A060734 1,2
%A A060734 _Frank Ellermann_, Apr 23 2001
%E A060734 Corrected by _Jeremy Gardiner_, Dec 26 2008