%I #24 Nov 29 2023 06:58:26

%S 1,2,4,5,3,9,10,6,8,16,17,11,7,15,25,26,18,12,14,24,36,37,27,19,13,23,

%T 35,49,50,38,28,20,22,34,48,64,65,51,39,29,21,33,47,63,81,82,66,52,40,

%U 30,32,46,62,80,100

%N Array of square numbers read by antidiagonals in up direction.

%C A simple permutation of natural numbers.

%C a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - _Boris Putievskiy_, Jan 09 2013

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>

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

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

%F a(n)=i^2-j+1 if i >= j, a(n)=(j-1)^2 + i if i < j, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Jan 09 2013

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

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

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

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

%t Table[ If[n < 2*k-1, k^2 + k - n, (n-k)^2 + k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 09 2013 *)

%o (Python)

%o t=int((math.sqrt(8*n-7) - 1)/ 2)

%o i=n-t*(t+1)/2

%o j=(t*t+3*t+4)/2-n

%o if i>=j:

%o result=i**2-j+1

%o else:

%o result=(j-1)**2+i

%o # _Boris Putievskiy_, Jan 09 2013

%Y Cf. A060734. Inverse permutation: A064788, the first inverse function (numbers of rows) A194258, the second inverse function (numbers of columns) A194195.

%K nonn,tabl

%O 1,2

%A _Frank Ellermann_, Apr 23 2001