A208234 Second inverse function (numbers of columns) for pairing function A188568.

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

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.

Formula

a(n) = -max(i,j)*((-1)^i-1)/2+min(i,j)*((-1)^i+1)/2, if i>=j

a(n) = max(i,j)*((-1)^j+1)/2-min(i,j)*((-1)^j-1)/2, if i<j,

where t = floor((-1+sqrt(8*n-7))/2), i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n.

Example

The start of the sequence as triangle array read by rows:
  1;
  2,1;
  1,2,3;
  4,2,3,1;
  1,4,3,2,5;
  6,2,4,3,5,1;
  1,6,3,4,5,2,7;
  ...
Row number k contains permutation numbers form 1 to k.

Prog

(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)//2
j=(t*t+3*t+4)//2-n
if i>=j:
    result=-max(i, j)*((-1)**i-1)/2+min(i, j)*((-1)**i+1)/2
else:
    result= max(i, j)*((-1)**j+1)/2-min(i, j)*((-1)**j-1)/2

Crossrefs

Cf. A188568.

Sequence in context: A374412 A363148 A054482 * A092543 A277083 A216222

Adjacent sequences: A208231 A208232 A208233 * A208235 A208236 A208237

Keyword

nonn

Author

Boris Putievskiy, Jan 10 2013

Status

approved