A213205 T(n,k) = ((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.

1, 5, 4, 2, 3, 6, 10, 9, 14, 13, 7, 8, 11, 12, 15, 19, 18, 23, 22, 27, 26, 16, 17, 20, 21, 24, 25, 28, 32, 31, 36, 35, 40, 39, 44, 43, 29, 30, 33, 34, 37, 38, 41, 42, 45, 49, 48, 53, 52, 57, 56, 61, 60, 65, 64, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 70

Offset: 1

Author

Boris Putievskiy, Feb 15 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). The order of the list:

T(1,1)=1;

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

. . .

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

. . .

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

Table contains:

row 1 accommodates elements A130883 in odd places,

row 2 is alternation of elements A100037 and A033816;

column 1 is alternation of elements A000384 and A091823,

column 2 is alternation of elements A014106 and A071355,

column 3 accommodates elements A130861 in even places;

main diagonal is alternation of elements A188135 and A033567,

diagonal 1, located above the main diagonal accommodates elements A033566 in even places,

diagonal 2, located above the main diagonal is alternation of elements A139271 and A024847,

diagonal 3, located above the main diagonal accommodates of elements A033585.

Examples

			The start of the sequence as table:
1....5...2..10...7..19..16...
4....3...9...8..18..17..31...
6...14..11..23..20..36..33...
13..12..22..21..35..34..52...
15..27..24..40..37..57..54...
26..25..39..38..56..55..77...
28..44..41..61..58..82..79...
. . .
The start of the sequence as triangle array read by rows:
1;
5,4;
2,3,6;
10,9,14,13;
7,8,11,12,15;
19,18,23,22,27,26;
16,17,20,21,24,25,28;
. . .
The start of the sequence as array read by rows, the length of row 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;
5,4,2,3,6;
10,9,14,13,7,8,11,12,15;
19,18,23,22,27,26,16,17,20,21,24,25,28;
. . .
Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+7, 2*r*r-5*r+6,...2*r*r-r-4, 2*r*r-r-3, 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. A211377, A130883, A100037, A033816, A000384, A091823, A014106, A071355, A130861, A188135, A033567, A033566, A139271, A024847, A033585, A002260, A004736, A003056, A003057.

Programs

Maple

T:=(n,k)->((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

Mathematica

T[n_, k_] := ((n+k)^2 - 4k + 3 + (-1)^k - 2(-1)^n - (n+k)(-1)^(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

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

Formula

As table

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

As linear sequence

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

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