A213196 Inverse permutation of A211377.

1, 4, 2, 3, 5, 6, 11, 7, 8, 12, 13, 9, 10, 14, 15, 22, 16, 17, 23, 24, 18, 19, 25, 26, 20, 21, 27, 28, 37, 29, 30, 38, 39, 31, 32, 40, 41, 33, 34, 42, 43, 35, 36, 44, 45, 56, 46, 47, 57, 58, 48, 49, 59, 60, 50, 51, 61, 62, 52, 53, 63, 64, 54, 55, 65, 66, 79

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

a(n)=(m1+m2-1)*(m1+m2-2)/2+m1, where

m1=(3*i+j-1-(-1)^i+(i+j-2)*(-1)*t)/4,

m2=((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,

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;
  4,2;
  3,5,6;
  11,7,8,12;
  13,9,10,14,15;
  22,16,17,23,24,18;
  19,25,26,20,21,27,28;
  . . .
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;
  4,2,3,5,6;
  11,7,8,12,13,9,10,14,15;
  22,16,17,23,24,18,19,25,26,20,21,27,28;
Row number r contains permutation of 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r: 2*r*r-3*r+2, 2*r*r-5*r+4, 2*r*r-5*r+5,... 2*r*r-r-1, 2*r*r-r.

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
m1=(3*i+j-1-(-1)**i+(i+j-2)*(-1)**(i+j))/4
m2=((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
result=(m1+m2-1)*(m1+m2-2)/2+m1

Crossrefs

Cf. A211377.

Sequence in context: A182272 A182273 A166016 * A275104 A010646 A284307

Adjacent sequences: A213193 A213194 A213195 * A213197 A213198 A213199

Keyword

nonn

Author

Boris Putievskiy, Mar 01 2013

Status

approved