%I #13 Feb 17 2022 14:17:59
%S 1,1,2,3,2,1,1,3,2,4,5,2,3,4,1,1,5,3,4,2,6,7,2,5,4,3,6,1,1,7,3,5,4,6,
%T 2,8,9,2,7,4,5,6,3,8,1,1,9,3,7,5,6,4,8,2,10,11,2,9,4,7,6,5,8,3,10,1
%N First inverse function (numbers of rows) for pairing function A188568.
%H Boris Putievskiy, <a href="/A208233/b208233.txt">Rows n = 1..140 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.
%F a(n) = max(i,j)*((-1)^i+1)/2-min(i,j)*((-1)^i-1)/2, if i>=j
%F a(n) = -max(i,j)*((-1)^j-1)/2+min(i,j)*((-1)^j+1)/2, if i<j,
%F where
%F t = floor((-1+sqrt(8*n-7))/2),
%F i = n-t*(t+1)/2,
%F j = (t*t+3*t+4)/2-n.
%e The start of the sequence as triangle array read by rows:
%e 1;
%e 1,2;
%e 3,2,1;
%e 1,3,2,4;
%e 5,2,3,41;
%e 1,5,3,4,2,6;
%e 7,2,5,4,3,6,1;
%e ...
%e Row number k contains permutation numbers form 1 to k.
%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= max(i,j)*((-1)**i+1)/2-min(i,j)*((-1)**i-1)/2
%o else:
%o result=-max(i,j)*((-1)**j-1)/2+min(i,j)*((-1)**j+1)/2
%Y Cf. A188568.
%K nonn,tabl
%O 1,3
%A _Boris Putievskiy_, Jan 10 2013