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

Author

Boris Putievskiy, Mar 01 2013

Keywords

• nonn

Examples

			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.
		

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

Crossrefs

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

Programs

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

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).