A208233 First inverse function (numbers of rows) for pairing function A188568.

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

Offset

1,3

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;
  1,2;
  3,2,1;
  1,3,2,4;
  5,2,3,41;
  1,5,3,4,2,6;
  7,2,5,4,3,6,1;
  ...
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: A339178 A026552 A333271 * A176270 A361802 A086437

Adjacent sequences: A208230 A208231 A208232 * A208234 A208235 A208236

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 10 2013

Status

approved