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A208233 First inverse function (numbers of rows) for pairing function A188568.

%I A208233 #13 Feb 17 2022 14:17:59
%S A208233 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 A208233 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 A208233 First inverse function (numbers of rows) for pairing function A188568.
%H A208233 Boris Putievskiy, <a href="/A208233/b208233.txt">Rows n = 1..140 of triangle, flattened</a>
%H A208233 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A208233 a(n) =  max(i,j)*((-1)^i+1)/2-min(i,j)*((-1)^i-1)/2, if i>=j
%F A208233 a(n) = -max(i,j)*((-1)^j-1)/2+min(i,j)*((-1)^j+1)/2, if i<j,
%F A208233 where
%F A208233 t = floor((-1+sqrt(8*n-7))/2),
%F A208233 i = n-t*(t+1)/2,
%F A208233 j = (t*t+3*t+4)/2-n.
%e A208233 The start of the sequence as triangle array read by rows:
%e A208233   1;
%e A208233   1,2;
%e A208233   3,2,1;
%e A208233   1,3,2,4;
%e A208233   5,2,3,41;
%e A208233   1,5,3,4,2,6;
%e A208233   7,2,5,4,3,6,1;
%e A208233   ...
%e A208233 Row number k contains permutation numbers form 1 to k.
%o A208233 (Python)
%o A208233 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A208233 i=n-t*(t+1)/2
%o A208233 j=(t*t+3*t+4)/2-n
%o A208233 if i>=j:
%o A208233    result= max(i,j)*((-1)**i+1)/2-min(i,j)*((-1)**i-1)/2
%o A208233 else:
%o A208233    result=-max(i,j)*((-1)**j-1)/2+min(i,j)*((-1)**j+1)/2
%Y A208233 Cf. A188568.
%K A208233 nonn,tabl
%O A208233 1,3
%A A208233 _Boris Putievskiy_, Jan 10 2013