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A209278 Second inverse function (numbers of rows) for pairing function A185180.

%I A209278 #29 Feb 16 2025 08:33:16
%S A209278 1,2,1,2,3,1,3,2,4,1,3,4,2,5,1,4,3,5,2,6,1,4,5,3,6,2,7,1,5,4,6,3,7,2,
%T A209278 8,1,5,6,4,7,3,8,2,9,1,6,5,7,4,8,3,9,2,10,1,6,7,5,8,4,9,3,10,2,11,1
%N A209278 Second inverse function (numbers of rows) for pairing function A185180.
%H A209278 Boris Putievskiy, <a href="/A209278/b209278.txt">Rows n = 1..140 of triangle, flattened</a>
%H A209278 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%H A209278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%F A209278 a(n) = floor((A003056(n)+3)/2) + floor(A002260(n)/2)*(-1)^(A002260(n)+A003056(n)).
%F A209278 a(n)= floor((t+3)/2)+ floor(i/2)*(-1)^(i+t),
%F A209278 where t=floor((-1+sqrt(8*n-7))/2), i=n-t*(t+1)/2.
%F A209278 T(r,2s-1)=s, T(r,2s)= r+s. (When read as square array by antidiagonals.)
%e A209278 The start of the sequence as table T(r,s) r,s >0 read by antidiagonals:
%e A209278 1...2...2...3...3...4...4...5...
%e A209278 1...3...2...4...3...5...4...6...
%e A209278 1...4...2...5...3...6...4...7...
%e A209278 1...5...2...6...3...7...4...8...
%e A209278 1...6...2...7...3...8...4...9...
%e A209278 1...7...2...8...3...9...4..10...
%e A209278 1...8...2...9...3..10...4..11...
%e A209278 . . .
%e A209278 The start of the sequence as triangle array read by rows:
%e A209278 1;
%e A209278 2, 1;
%e A209278 2, 3, 1;
%e A209278 3, 2, 4, 1;
%e A209278 3, 4, 2, 5, 1;
%e A209278 4, 3, 5, 2, 6, 1;
%e A209278 4, 5, 3, 6, 2, 7, 1;
%e A209278 5, 4, 6, 3, 7, 2, 8, 1;
%e A209278 . . .
%e A209278 Row number r contains permutation numbers form 1 to r.
%e A209278 If r is odd (r+1)/2, (r+1)/2 +1, (r+1)/2 -1, ... 2, r, 1.
%e A209278 If r is even r/2 + 1, r/2, r/2 + 2, ...  2, r, 1.
%t A209278 T[r_, s_] := If[OddQ[s], (s+1)/2, r + s/2];
%t A209278 Table[T[r-s+1, s], {r, 1, 11}, {s, r, 1, -1}] // Flatten (* _Jean-François Alcover_, Nov 19 2019 *)
%o A209278 (Python)
%o A209278 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A209278 i=n-t*(t+1)/2
%o A209278 result=int((t+3)/2)+int(i/2)*(-1)**(i+t)
%o A209278 (PARI) T(r,s)=s\2+if(bittest(s,0),1,r) \\ - _M. F. Hasler_, Jan 15 2013
%Y A209278 Cf. A185180, A092542, A092543.
%K A209278 nonn,tabl
%O A209278 1,2
%A A209278 _Boris Putievskiy_, Jan 15 2013