A216249 T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2-2, 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, 3, 2, 4, 5, 6, 8, 7, 12, 11, 9, 10, 13, 14, 15, 17, 16, 21, 20, 25, 24, 18, 19, 22, 23, 26, 27, 28, 30, 29, 34, 33, 38, 37, 42, 41, 31, 32, 35, 36, 39, 40, 43, 44, 45, 47, 46, 51, 50, 55, 54, 59, 58, 63, 62, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 66, 68, 67, 72, 71, 76, 75, 80, 79, 84, 83, 88, 87

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

. . .

T(2,2*m-1), T(1,2*m), T(1,2*m+1), T(2,2*m), T(2*m-3,4), ... T(2*m,1), T(2*m-1,2), T(2*m-1,3), T(2*m,2), T(2*m+1,1);

. . .

Movement along two adjacent antidiagonals - step to the northeast, step to the east, step to the southwest, 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   3  4    8   9  17  18...
   2   5  7   10  16  19  29...
   6  12  13  21  22  34  35...
  11  14  20  23  33  36  50...
  15  25  26  38  39  55  56...
  24  27  37  40  54  57  75...
  28  42  43  59  60  80  81...
  ...
The start of the sequence as triangular array read by rows:
   1;
   3,  2;
   4,  5,  6;
   8,  7, 12, 11;
   9, 10, 13, 14, 15;
  17, 16, 21, 20, 25, 24;
  18, 19, 22, 23, 26, 27, 28;
  ...
As an array read by rows, where the length of row number r is 4*r-3:
First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
  1;
  3,   2,   4,   5,   6;
  8,   7,  12,  11,   9,  10,  13,  14,  15;
  17, 16,  21,  20,  25,  24,  18,  19,  22,  23,  26,  27,  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+5, 2*r*r-5*r+4, ...2*r*r-r-1, 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 A100037, A033816, A130883, A100039, A100038; in columns A000384, A071355, A091823, A014106.

Programs

Mathematica

T[n_, k_] := ((n+k)^2 - 4k + 3 - 2(-1)^n + (-1)^k - (n+k-4)(-1)^(n+k))/2 - 2Boole[k == 1 && OddQ[n]];
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 20 2019 *)

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-2

Formula

As a table:

T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2-2, 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 a linear sequence:

a(n) = ((t+2)^2-4*j+3-2*(-1)^i+(-1)^j-(t-2)*(-1)^t)/2-2, 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).