A210535 Second inverse function (numbers of columns) for pairing function A209293.

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

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

Formula

a(n) = 2*A200260(n)-A101688(n)*(4*A002260(n)-2*A003056(n)-3).

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

Example

The start of the sequence as triangle array read by rows:
  1;
  2,1;
  2,3,1;
  2,4,3,1;
  2,4,5,3,1;
  2,4,6,5,3,1;
  2,4,6,7,5,3,1;
  2,4,6,8,7,5,3,1;
  . . .
Row number r contains permutation numbers from 1 to r: 2,4,6,...2*floor(r/2),2*floor(r/2)-1,2*floor(r/2)-3,...3,1.

Prog

(Python)
t=int((math.sqrt(8*n-7)-1)/2)
i=n-t*(t+1)/2
v=int((2*n+1-t*(t+1))/(t+3))
result=2*i-v*(4*i-2*t-3)

Crossrefs

Cf. A209293, A200260, A101688, A003056, A220073.

Sequence in context: A376308 A194976 A195082 * A194983 A195073 A194987

Adjacent sequences: A210532 A210533 A210534 * A210536 A210537 A210538

Keyword

nonn

Author

Boris Putievskiy, Jan 28 2013

Status

approved