A214929 A209293 as table read layer by layer - layer clockwise, layer counterclockwise and so on.

1, 3, 4, 2, 5, 9, 14, 7, 6, 10, 11, 20, 23, 17, 12, 8, 13, 19, 26, 34, 43, 30, 27, 16, 15, 21, 22, 35, 38, 53, 58, 48, 39, 31, 24, 18, 25, 33, 42, 52, 63, 75, 88, 69, 64, 47, 44, 29, 28, 36, 37, 54, 57, 76, 81, 102, 109, 95, 82, 70, 59, 49, 40, 32, 41, 51, 62

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.

Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Table read by boustrophedonic ("ox-plowing") method. Let m be natural number. The order of the list:

T(1,1)=1;

T(2,1), T(2,2), T(1,2);

...

T(1,2*m+1), T(2,2*m+1), ... T(2*m,2*m+1), T(2*m+1,2*m+1), T(2*m+1,2*m), ... T(2*m+1,1);

T(2*m,1), T(2*m,2), ... T(2*m,2*m-1), T(2*m,2*m), T(2*m-1,2*m), ... T(1,2*m);

...

The first row is layer read clockwise, the second row is layer counterclockwise.

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 Function.

Index entries for sequences that are permutations of the natural numbers

Formula

As table

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

where i = (t mod 2)*min(t; n-(t-1)^2) + (t+1 mod 2)*min(t; t^2-n+1), j = (t mod 2)*min(t; t^2-n+1) + (t+1 mod 2)*min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1.

Example

The start of the sequence as table:
   1   2   5   8  13  18...
   3   4   9  12  19  24...
   6   7  14  17  26  31...
  10  11  20  23  34  39...
  15  16  27  30  43  48...
  21  22  35  38  53  58...
  ...
The start of the sequence as triangle array read by rows:
  1;
  3,4,2;
  5,9,14,7,6;
  10,11,20,23,17,12,8;
  13,19,26,34,43,30,27,16,15;
  21,22,35,38,53,58,48,39,31,24,18;
  ...
Row number r contains 2*r-1 numbers.

Prog

(Python)
import math
def a(n):
    t=math.isqrt(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)
    m1=((i+j)//2)+(i//2)*(-1)**(2*i+j-1)
    m2=((i+j+1)//2)+(i//2)*(-1)**(2*i+j-2)
    return (m1+m2-1)*(m1+m2-2)//2+m1

Crossrefs

Cf. A081344, A209293, A209279, A209278, A185180; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

Sequence in context: A276958 A275848 A367651 * A205152 A162196 A179297

Adjacent sequences: A214926 A214927 A214928 * A214930 A214931 A214932

Keyword

nonn,tabf,easy

Author

Boris Putievskiy, Mar 11 2013

Status

approved