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

1,2

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 numder 1 A000982,

row number 2 A097063.

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

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.

Example

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;

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 *)

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=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

Crossrefs

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

Sequence in context: A073291 A073280 A073884 * A328021 A361504 A377157

Adjacent sequences: A209290 A209291 A209292 * A209294 A209295 A209296

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 16 2013

Status

approved