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%I #44 Jul 26 2023 14:32:35
%S 1,1,2,2,1,3,2,3,1,4,3,2,4,1,5,3,4,2,5,1,6,4,3,5,2,6,1,7,4,5,3,6,2,7,
%T 1,8,5,4,6,3,7,2,8,1,9,5,6,4,7,3,8,2,9,1,10,6,5,7,4,8,3,9,2,10,1,11,6,
%U 7,5,8,4,9,3,10,2,11,1,12,7,6,8,5,9,4,10,3,11,2,12,1,13
%N First inverse function (numbers of rows) for pairing function A185180.
%C The triangle equals A158946 with the first column removed. - _Georg Fischer_, Jul 26 2023
%H Boris Putievskiy, <a href="/A209279/b209279.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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%F a(n) = floor((A003056(n)+2)/2)+ floor(A002260(n)/2)*(-1)^(A002260(n)+A003056(n)+1).
%F a(n) = |A128180(n)|.
%F a(n) = floor((t+2)/2) + floor(i/2)*(-1)^(i+t+1), where t=floor((-1+sqrt(8*n-7))/2), i=n-t*(t+1)/2.
%F T(r,2s)=s, T(r,2s-1)= r+s-1.(When read as table T(r,s) by antidiagonals.)
%F T(n,k) = ceiling((n + (-1)^(n-k)*k)/2) = (n+k)/2 if n-k even, otherwise (n-k+1)/2. - _M. F. Hasler_, May 30 2020
%e The start of the sequence as table T(r,s) r,s >0 read by antidiagonals:
%e 1...1...2...2...3...3...4...4...
%e 2...1...3...2...4...3...5...4...
%e 3...1...4...2...5...3...6...4...
%e 4...1...5...2...6...3...7...4...
%e 5...1...6...2...7...3...8...4...
%e 6...1...7...2...8...3...9...4...
%e 7...1...8...2...9...3..10...4...
%e ...
%e The start of the sequence as triangle array read by rows:
%e 1;
%e 1, 2;
%e 2, 1, 3;
%e 2, 3, 1, 4;
%e 3, 2, 4, 1, 5;
%e 3, 4, 2, 5, 1, 6;
%e 4, 3, 5, 2, 6, 1, 7;
%e 4, 5, 3, 6, 2, 7, 1, 8;
%e ...
%e Row number r contains permutation numbers form 1 to r.
%e If r is odd (r+1)/2, (r+1)/2-1, (r+1)/2+1,...r-1, 1, r.
%e If r is even r/2, r/2+1, r/2-1, ... r-1, 1, r.
%t T[n_, k_] := Abs[(2*k - 1 + (-1)^(n - k)*(2*n + 1))/4];
%t Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 14 2018, after _Andrew Howroyd_ *)
%o (PARI) T(n, k)=abs((2*k-1+(-1)^(n-k)*(2*n+1))/4) \\ _Andrew Howroyd_, Dec 31 2017
%o (Python) # Edited by _M. F. Hasler_, May 30 2020
%o def a(n):
%o t = int((math.sqrt(8*n-7) - 1)/2);
%o i = n-t*(t+1)/2;
%o return int(t/2)+1+int(i/2)*(-1)**(i+t+1)
%Y Cf. A158946, A185180, A128180, A092542, A092543, A209278.
%K nonn,tabl
%O 1,3
%A _Boris Putievskiy_, Jan 15 2013
%E Data corrected by _Andrew Howroyd_, Dec 31 2017
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