A213195 Second inverse function (of columns) for pairing function A211377.

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

Offset

1,2

Links

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Eric Weisstein's World of Mathematics, Pairing functions

Index entries for sequences that are permutations of the natural numbers

Formula

See Python program.

Example

The start of the sequence as triangle array read by rows:
  1;
  3,2;
  1,2,1;
  5,4,3,4;
  3,2,1,2,1;
  7,6,5,6,5,4;
  3,4,3,2,1,2,1;
  . . .
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of above triangle array.
Last  2*r-1 numbers are from the row number 2*r-1 of above triangle array.
  1;
  3,2,1,2,1;
  5,4,3,4,3,2,1,2,1;
  7,6,5,6,5,4,3,4,3,2,1,2,1;
  ...
Row number r is 2*r-1, 2*r-2, 2*r-3, 2*r-2, {row number r-1}.

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
result=((1+(-1)**i)*((1+(-1)**j)*2*int((j+2)/4)-(-1+(-1)**j)*(2*int((i+4)/4)+2*int(j/2)))-(-1+(-1)**i)*((1+(-1)**j)*(1+2*int(i/4)+2*int(j/2))-(-1+(-1)**j)*(1+2*int(j/4))))/4

Crossrefs

Cf. A211377.

Sequence in context: A096248 A079109 A079099 * A256794 A366899 A068929

Adjacent sequences: A213192 A213193 A213194 * A213196 A213197 A213198

Keyword

nonn,tabl

Author

Boris Putievskiy, Mar 01 2013

Status

approved