A214928 A209293 as table read layer by layer clockwise.

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

Offset: 1

Author

Boris Putievskiy, Mar 11 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.

Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). The order of the list:

T(1,1)=1;

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

. . .

T(1,n), T(2,n), ... T(n-1,n), T(n,n), T(n,n-1), ... T(n,1);

. . .

Examples

			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;
  2,4,3;
  5,9,14,7,6;
  8,12,17,23,20,11,10;
  13,19,26,34,43,30,27,16,15;
  18,24,31,39,48,58,53,38,35,22,21;
  . . .
Row number r contains 2*r-1 numbers.
		

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. A209293, A209279, A209278, A185180, A060734, A060736; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

Programs

Python

t=int((math.sqrt(n-1)))+1
i=min(t,n-(t-1)**2)
j=min(t,t**2-n+1)
m1=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
result=(m1+m2-1)*(m1+m2-2)/2+m1

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