A209293 Inverse permutation of A185180.

1, 2, 3, 5, 4, 6, 8, 9, 7, 10, 13, 12, 14, 11, 15, 18, 19, 17, 20, 16, 21, 25, 24, 26, 23, 27, 22, 28, 32, 33, 31, 34, 30, 35, 29, 36, 41, 40, 42, 39, 43, 38, 44, 37, 45, 50, 51, 49, 52, 48, 53, 47, 54, 46, 55, 61, 60, 62, 59, 63, 58, 64, 57, 65, 56, 66, 72, 73, 71, 74, 70, 75, 69, 76, 68, 77, 67

Offset: 1

Author

Boris Putievskiy, Jan 16 2013

Keywords

• nonn
• tabl

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.

Enumeration table T(n,k) by diagonals. The order of the list

if n is odd - T(n-1,2),T(n-3,4),...,T(2,n-1),T(1,n),T(3,n-2),...T(n,1).

if n is even - T(n-1,2),T(n-3,4),...,T(3,n-2),T(1,n),T(2,n-1),...T(n,1).

Table T(n,k) contains:

Column number 1 A000217,

column number 2 A000124,

column number 3 A000096,

column number 4 A152948,

column number 5 A034856,

column number 6 A152950,

column number 7 A055998.

Row number 1 A000982,

row number 2 A097063.

Examples

			The start of the sequence as table:
  1....2...5...8..13..18...25...32...41...
  3....4...9..12..19..24...33...40...51...
  6....7..14..17..26..31...42...49...62...
  10..11..20..23..34..39...52...59...74...
  15..16..27..30..43..48...63...70...87...
  21..22..35..38..53..58...75...82..101...
  28..29..44..47..64..69...88...95..116...
  36..37..54..57..76..81..102..109..132...
  45..46..65..68..89..94..117..124..149...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  2,3;
  5,4,6;
  8,9,7,10;
  13,12,14,11,15;
  18,19,17,20,16,21;
  25,24,26,23,27,22,28;
  32,33,31,34,30,35,29,36;
  41,40,42,39,43,38,44,37,45;
  . . .
Row number r contains permutation from r numbers:
if r is odd  ceiling(r^2/2), ceiling(r^2/2)+1, ceiling(r^2/2)-1, ceiling(r^2/2)+2, ceiling(r^2/2)-2,...r*(r+1)/2;
if r is even ceiling(r^2/2), ceiling(r^2/2)-1, ceiling(r^2/2)+1, ceiling(r^2/2)-2, ceiling(r^2/2)+2,...r*(r+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

Crossrefs

Cf. A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

Programs

Mathematica

max = 10; row[n_] := Table[Ceiling[(n + k - 1)^2/2] + If[OddQ[k], 1, -1]*Floor[n/2], {k, 1, max}]; t = Table[row[n], {n, 1, max}]; Table[t[[n - k + 1, k]], {n, 1, max}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 17 2013 *)

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=int((i+j)/2)+int(i/2)*(-1)**(i+t+1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(i+t)
m=(m1+m2-1)*(m1+m2-2)/2+m1

Formula

As table T(n,k) read by antidiagonals

T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.

As linear sequence

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

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

m2 = int((i+j+1)/2)+int(i/2)*(-1)^(i+t),

t = int((math.sqrt(8*n-7) - 1)/ 2),

i = n-t*(t+1)/2,

j = (t*t+3*t+4)/2-n.