A194280 Inverse permutation to A081344.

1, 2, 5, 3, 6, 9, 13, 8, 4, 7, 12, 18, 25, 19, 14, 10, 15, 20, 26, 33, 41, 32, 24, 17, 11, 16, 23, 31, 40, 50, 61, 51, 42, 34, 27, 21, 28, 35, 43, 52, 62, 73, 85, 72, 60, 49, 39, 30, 22, 29, 38, 48, 59, 71, 84, 98, 113

Offset: 1

Author

Boris Putievskiy, Dec 23 2012

Keywords

• nonn

Comments

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

Call a "layer" a pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). This sequence is A188568 as table read layer by layer clockwise.

The same table A188568 read by boustrophedon ("ox-plowing") method - layer clockwise, layer counterclockwise and so on - is A064790. - Boris Putievskiy, Mar 14 2013

Examples

			From _Boris Putievskiy_, Mar 14 2013: (Start)
The start of the sequence as table:
  1....2...6...7..15..16..28...
  3....5...9..12..20..23..35...
  4....8..13..18..26..31..43...
  10..14..19..25..33..40..52...
  11..17..24..32..41..50..62...
  21..27..34..42..51..61..73...
  22..30..39..49..60..72..85...
  ...
The start of the sequence as triangular array read by rows:
  1;
  2,5,3;
  6,9,13,8,4;
  7,12,18,25,19,14,10;
  15,20,26,33,41,32,24,17,11;
  16,23,31,40,50,61,51,42,34,27,21;
  28,35,43,52,62,73,85,72,60,49,39,30,22;
  ...
Row number r contains 2*r-1 numbers. (End)
		

Links

• Boris Putievskiy, Rows n = 1..140 of triangle, flattened
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A081344, A064790, A188568.

Programs

Python

t=int(math.sqrt(n-1))+1
i=(t % 2)*min(t,n-(t-1)**2) + ((t+1) % 2)*min(t,t**2-n+1)
j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)
m=(i+j-1)*(i+j-2)/2+j

Formula

a(n) = (i+j-1)*(i+j-2)/2+j, where

i = mod(t;2)*min{t; n - (t - 1)^2} + mod(t + 1; 2)*min{t; t^2 - n + 1}

j = mod(t;2)*min{t; t^2 - n + 1} + mod(t + 1; 2)*min{t; n - (t - 1)^2},

t = int(math.sqrt(n-1))+1.