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

1,2

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.

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

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

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.

Example

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)

Prog

(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

Crossrefs

Cf. A081344, A064790, A188568.

Sequence in context: A359008 A239970 A111202 * A163362 A243061 A242911

Adjacent sequences: A194277 A194278 A194279 * A194281 A194282 A194283

Keyword

nonn

Author

Boris Putievskiy, Dec 23 2012

Status

approved