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A208234 Second inverse function (numbers of columns) for pairing function A188568.

%I A208234 #21 Feb 16 2022 11:59:10
%S A208234 1,2,1,1,2,3,4,2,3,1,1,4,3,2,5,6,2,4,3,5,1,1,6,3,4,5,2,7,8,2,6,4,5,3,
%T A208234 7,1,1,8,3,6,5,4,7,2,9,10,2,8,4,6,5,7,3,9,1,1,10,3,8,5,6,7,4,9,2,11
%N A208234 Second inverse function (numbers of columns) for pairing function A188568.
%H A208234 Boris Putievskiy, <a href="/A208234/b208234.txt">Rows n = 1..140 of triangle, flattened</a>
%H A208234 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A208234 a(n) = -max(i,j)*((-1)^i-1)/2+min(i,j)*((-1)^i+1)/2,  if i>=j
%F A208234 a(n) =  max(i,j)*((-1)^j+1)/2-min(i,j)*((-1)^j-1)/2,  if i<j,
%F A208234 where t = floor((-1+sqrt(8*n-7))/2), i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n.
%e A208234 The start of the sequence as triangle array read by rows:
%e A208234   1;
%e A208234   2,1;
%e A208234   1,2,3;
%e A208234   4,2,3,1;
%e A208234   1,4,3,2,5;
%e A208234   6,2,4,3,5,1;
%e A208234   1,6,3,4,5,2,7;
%e A208234   ...
%e A208234 Row number k contains permutation numbers form 1 to k.
%o A208234 (Python)
%o A208234 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A208234 i=n-t*(t+1)//2
%o A208234 j=(t*t+3*t+4)//2-n
%o A208234 if i>=j:
%o A208234     result=-max(i,j)*((-1)**i-1)/2+min(i,j)*((-1)**i+1)/2
%o A208234 else:
%o A208234     result= max(i,j)*((-1)**j+1)/2-min(i,j)*((-1)**j-1)/2
%Y A208234 Cf. A188568.
%K A208234 nonn
%O A208234 1,2
%A A208234 _Boris Putievskiy_, Jan 10 2013