A213194 First inverse function (numbers of rows) for pairing function A211377.

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

Offset

1,4

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

a(n) = (3*A002600(n)+A004736(n)-1-(-1)^A002260(n)+A003056(n)*(-1)^A003057(n))/4;

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

Example

The start of the sequence as triangle array read by rows:
  1;
  1,1;
  2,2,3;
  1,1,2,2;
  3,3,4,4,5;
  1,1,2,2,3,3;
  4,4,5,5,6,6,7;
  1,1,2,2,3,3,4,4;
  5,5,6,6,7,7,8,8,9;
  1,1,2,2,3,3,4,4,5,5;
  . . .
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;
  1,1,2,2,3;
  1,1,2,2,3,3,4,4,5;
  1,1,2,2,3,3,4,4,5,5,6,6,7;
  1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9;
  . . .
Row r contains numbers 1,2,3,...2*r-2 repeated twice, row ends 2*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=(3*i+j-1-(-1)**i+(i+j-2)*(-1)**(i+j))/4

Crossrefs

Cf. A211377, A002260, A004736, A003056, A003057.

Sequence in context: A074313 A367438 A303506 * A208970 A213939 A267485

Adjacent sequences: A213191 A213192 A213193 * A213195 A213196 A213197

Keyword

nonn

Author

Boris Putievskiy, Mar 01 2013

Status

approved