A216250 T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1; T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

1, 2, 3, 5, 4, 6, 7, 8, 11, 12, 10, 9, 14, 13, 15, 16, 17, 20, 21, 24, 25, 19, 18, 23, 22, 27, 26, 28, 29, 30, 33, 34, 37, 38, 41, 42, 32, 31, 36, 35, 40, 39, 44, 43, 45, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 49, 48, 53, 52, 57, 56, 61, 60, 65, 64, 66, 67

Offset: 1

Author

Boris Putievskiy, Mar 14 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.

Enumeration table T(n,k). Let m be natural number. The order of the list:

T(1,1)=1;

T(1,2), T(2,1), T(2,2), T(1,3), T(3,1);

. . .

T(1,2*m), T(2,2*m-1), T(2,2*m), T(1,2*m+1), T(3,2*m-2), ... T(2*m-1,2), T(2*m,1), T(2*m,2), T(2*m-1,3), T(2*m+1,1);

. . .

Movement along two adjacent antidiagonals - step to the southwest, step east, step to the northeast, 3 steps to the west, 2 steps to the south and so on. The length of each step is 1.

Examples

			The start of the sequence as table:
  1....2...5...7..10..16..19...
  3....4...8...9..17..18..30...
  6...11..14..20..23..33..36...
  12..13..21..22..34..35..51...
  15..24..27..37..40..54..57...
  25..26..38..39..55..56..76...
  28..41..44..58..61..79..82...
  . . .
The start of the sequence as triangular array read by rows:
  1;
  2,3;
  5,4,6;
  7,8,11,12;
  10,9,14,13,15;
  16,17,20,21,24,25;
  19,18,23,22,27,26,28;
  . . .
The start of the sequence as array read by rows, with length of row r: 4*r-3:
First 2*r-2 numbers are from the row number 2*r-2 of above triangle array.
Last  2*r-1 numbers are from the row number 2*r-1 of above triangle array.
  1;
  2,3,5,4,6;
  7,8,11,12,10,9,14,13,15;
  16,17,20,21,24,25,19,18,23,22,27,26,28;
  . . .
Row number r contains permutation of the 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r: 2*r*r-5*r+4, 2*r*r-5*r+5, ...2*r*r-r-2, 2*r*r-r.
		

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. A213205, A213171, A213197, A210521; table T(n,k) contains: in rows A130883, A033816, A100037, A100038, A100039; in columns A000384, A014106, A071355, A091823, A130861.

Programs

Python

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-4*j+3-(-1)**j-2*(-1)**i-(t-2)*(-1)**t)/2
if j==1 and (i%2)==1:
   result=result-3

Formula

As table

T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1;

T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2, else.

As linear sequence

a(n) = ((t+2)^2-4*j+3-2*(-1)^i-(-1)^j-(t-2)*(-1)^t)/2-3, if j=1 and (i mod 2)=1;

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