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

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, 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

Offset: 1

Author

Boris Putievskiy, Jan 15 2013

Keywords

• nonn
• tabl

Examples

			The start of the sequence as table T(r,s) r,s >0 read by antidiagonals:
1...2...2...3...3...4...4...5...
1...3...2...4...3...5...4...6...
1...4...2...5...3...6...4...7...
1...5...2...6...3...7...4...8...
1...6...2...7...3...8...4...9...
1...7...2...8...3...9...4..10...
1...8...2...9...3..10...4..11...
. . .
The start of the sequence as triangle array read by rows:
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, 8, 1;
. . .
Row number r contains permutation numbers form 1 to r.
If r is odd (r+1)/2, (r+1)/2 +1, (r+1)/2 -1, ... 2, r, 1.
If r is even r/2 + 1, r/2, r/2 + 2, ...  2, r, 1.
		

Links

• Boris Putievskiy, Rows n = 1..140 of triangle, flattened
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
• Eric Weisstein's World of Mathematics, Pairing functions

Crossrefs

Cf. A185180, A092542, A092543.

Programs

Mathematica

T[r_, s_] := If[OddQ[s], (s+1)/2, r + s/2];
Table[T[r-s+1, s], {r, 1, 11}, {s, r, 1, -1}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

PARI

T(r,s)=s\2+if(bittest(s,0),1,r) \\ - M. F. Hasler, Jan 15 2013

Python

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
result=int((t+3)/2)+int(i/2)*(-1)**(i+t)

Formula

a(n) = floor((A003056(n)+3)/2) + floor(A002260(n)/2)*(-1)^(A002260(n)+A003056(n)).

a(n)= floor((t+3)/2)+ floor(i/2)*(-1)^(i+t),

where t=floor((-1+sqrt(8*n-7))/2), i=n-t*(t+1)/2.

T(r,2s-1)=s, T(r,2s)= r+s. (When read as square array by antidiagonals.)