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

1,2

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

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.)

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A185180, A092542, A092543.

Sequence in context: A344084 A249783 A352964 * A326921 A370408 A358090

Adjacent sequences: A209275 A209276 A209277 * A209279 A209280 A209281

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 15 2013

Status

approved