A221216 T(n,k) = ((n+k)^2-2*(n+k)+4-(3*n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

1, 5, 6, 4, 3, 2, 12, 13, 14, 15, 11, 10, 9, 8, 7, 23, 24, 25, 26, 27, 28, 22, 21, 20, 19, 18, 17, 16, 38, 39, 40, 41, 42, 43, 44, 45, 37, 36, 35, 34, 33, 32, 31, 30, 29, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 80

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.

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

T(1,1)=1;

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

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

. . .

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

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

. . .

First row contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.

Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)}, read downwards.

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

As table

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

As linear sequence

a(n) = (A003057(n)^2-2*A003057(n)+4-(3*A002260(n)+A004736(n)-2)*(-1)^A003056(n))/2; a(n) = ((t+2)^2-2*(t+2)+4-(i+3*j-2)*(-1)^t)/2,

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:
  1....5...4..12..11..23..22...
  6....3..13..10..24..21..39...
  2...14...9..25..20..40..35...
  15...8..26..19..41..34..60...
  7...27..18..42..33..61..52...
  28..17..43..32..62..51..85...
  16..44..31..63..50..86..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  5,6;
  4,3,2;
  12,13,14,15;
  11,10,9,8,7;
  23,24,25,26,27,28;
  22,21,20,19,18,17,16;
  . . .
Row number r consecutive contains r numbers.
If r is odd,  row is decreasing.
If r is even, row is increasing.

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
result=((t+2)**2-2*(t+2)+4-(3*i+j-2)*(-1)**t)/2

Crossrefs

Cf. A211394, A221215, A002260, A004736, A003057;

table T(n,k) contains: in rows A084849, A096376, A014105, A014107, A168244, A033537, A100040, A100041;

in columns A130883, A000384, A014106, A033816, A100037, A091823, A071355, A100038, A100039,A130861;

main diagonal and parallel diagonals are A058331, A001844, A005893,A046092, A093328, A142463, A090288, A059993, A051890, A001105, A097080, A056220, A137882, A054000.

Sequence in context: A099038 A064206 A087197 * A229073 A019818 A187146

Adjacent sequences: A221213 A221214 A221215 * A221217 A221218 A221219

Keyword

nonn,tabl

Author

Boris Putievskiy, Feb 22 2013

Status

approved