A209279 First inverse function (numbers of rows) for pairing function A185180.

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, 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, 7, 5, 8, 4, 9, 3, 10, 2, 11, 1, 12, 7, 6, 8, 5, 9, 4, 10, 3, 11, 2, 12, 1, 13

Offset

1,3

Comments

The triangle equals A158946 with the first column removed. - Georg Fischer, Jul 26 2023

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

a(n) = |A128180(n)|.

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.

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

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

Example

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

Mathematica

T[n_, k_] := Abs[(2*k - 1 + (-1)^(n - k)*(2*n + 1))/4];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

Prog

(PARI) T(n, k)=abs((2*k-1+(-1)^(n-k)*(2*n+1))/4) \\ Andrew Howroyd, Dec 31 2017
(Python) # Edited by M. F. Hasler, May 30 2020
def a(n):
   t = int((math.sqrt(8*n-7) - 1)/2);
   i = n-t*(t+1)/2;
   return int(t/2)+1+int(i/2)*(-1)**(i+t+1)

Crossrefs

Cf. A158946, A185180, A128180, A092542, A092543, A209278.

Sequence in context: A002947 A241605 A128180 * A074754 A322529 A349526

Adjacent sequences: A209276 A209277 A209278 * A209280 A209281 A209282

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 15 2013

Extensions

Data corrected by Andrew Howroyd, Dec 31 2017

Status

approved