A213928 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.

1, 4, 2, 5, 3, 9, 16, 6, 8, 10, 25, 15, 7, 11, 17, 26, 24, 14, 12, 18, 36, 49, 27, 23, 13, 19, 35, 37, 64, 48, 28, 22, 20, 34, 38, 50, 65, 63, 47, 29, 21, 33, 39, 51, 81, 100, 66, 62, 46, 30, 32, 40, 52, 80, 82, 121, 99, 67, 61, 45, 31, 41, 53, 79, 83, 101

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.In general, let b(z) be a sequence of integer 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). Natural numbers placed in table T(n,k) layer by layer. The order of placement - layer is counterclockwise, if b(z) is odd; layer is clockwise if b(z) is even. T(n,k) read by antidiagonals.For A219159 - the order of the placement - at the beginning m layers counterclockwise, next m layers clockwise and so on - b(z)=floor((z-1)/m)+1. For this sequence b(z)=z^2 mod 3.

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

For general case.

As table

T(n,k) = ((1+(-1)^(b(k)-1))*(k^2-n+1)-(-1+(-1)^(b(k)-1))*((k-1)^2 +n))/2, if k >= n;

T(n,k) = ((1+(-1)^b(n))*(n^2-k+1)-(-1+(-1)^b(n))*((n-1)^2 +k))/2, if n >k.

As linear sequence

a(n) = ((1+(-1)^(b(j)-1))*(j^2-i+1)-(-1+(-1)^(b(j)-1))*((j-1)^2 +i))/2, if j >= i;

a(n) = ((1+(-1)^b(i))*(i^2-j+1)-(-1+(-1)^b(i))*((i-1)^2 +j))/2, if i >j;

where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

For this sequence b(z)=z^2 mod 3.

As table

T(n,k) = ((1+(-1)^(k^2 mod 3-1))*(k^2-n+1)-(-1+(-1)^(k^2 mod 3-1))*((k-1)^2 +n))/2, if k >= n;

T(n,k) = ((1+(-1)^(n^2 mod 3))*(n^2-k+1)-(-1+(-1)^(n^2 mod 3))*((n-1)^2 +k))/2, if n >k.

As linear sequence

a(n) = ((1+(-1)^(j^2 mod 3-1))*(j^2-i+1)-(-1+(-1)^(j^2 mod 3-1))*((j-1)^2 +i))/2, if j >= i;

a(n) = ((1+(-1)^(i^2 mod 3))*(i^2-j+1)-(-1+(-1)^(i^2 mod 3))*((i-1)^2 +j))/2, if i >j;

where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

Example

The start of the sequence as table.
The direction of the placement denotes by ">" and  "v".
  ..........v...........v...........v
  >1....4...5..16..25..26..49..64..65...
  >2....3...6..15..24..27..48..63..66...
  .9....8...7..14..23..28..47..62..67...
  >10..11..12..13..22..29..46..61..68...
  >17..18..19..20..21..30..45..60..69...
  .36..35..34..33..32..31..44..59..70...
  >37..38..39..40..41..42..43..58..71...
  >50..51..52..53..54..55..56..57..72...
  .81..80..79..78..77..76..75..74..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  4,2;
  5,3,9;
  16,6,8,10;
  25,15,7,11,17;
  26,24,14,12,18,36;
  49,27,23,13,19,35,37;
  64,48,28,22,20,34,38,50;
  65,63,47,29,21,33,39,51,81;
  . . .

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
if j>=i:
   result=((1+(-1)**(j**2%3-1))*(j**2-i+1)-(-1+(-1)**(j**2%3-1))*((j-1)**2 +i))/2
else:
   result=((1+(-1)**(i**2%3))*(i**2-j+1)-(-1+(-1)**(i**2%3))*((i-1)**2 +j))/2

Crossrefs

Cf. A219159, A081344, A194280, A042964, A130196, A011655, A220516.

Sequence in context: A218035 A090964 A219159 * A065189 A165275 A371192

Adjacent sequences: A213925 A213926 A213927 * A213929 A213930 A213931

Keyword

nonn,tabl

Author

Boris Putievskiy, Mar 06 2013

Status

approved