A221215 T(n,k)= ((n+k)^2-2*(n+k)+4-(n+3*k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

1, 6, 5, 2, 3, 4, 15, 14, 13, 12, 7, 8, 9, 10, 11, 28, 27, 26, 25, 24, 23, 16, 17, 18, 19, 20, 21, 22, 45, 44, 43, 42, 41, 40, 39, 38, 29, 30, 31, 32, 33, 34, 35, 36, 37, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 91

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

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

. . .

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

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

. . .

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

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

The same as A211394, except for reversed order in even diagonals. - M. F. Hasler, Feb 26 2013

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-(n+3*k-2)*(-1)^(n+k))/2.

As linear sequence

a(n) = (A003057(n)^2-2*A003057(n)+4-(A002260(n)+3*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....6...2..15...7..28..16...
5....3..14...8..27..17..44...
4...13...9..26..18..43..31...
12..10..25..19..42..32..63...
11..24..20..41..33..62..50...
23..21..40..34..61..51..86...
22..39..35..60..52..85..73...
. . .
The start of the sequence as triangle array read by rows:
1;
6,5;
2,3,4;
15,14,13,12;
7,8,9,10,11;
28,27,26,25,24,23;
16,17,18,19,20,21,22;
. . .
Row number r contains r consecutive numbers.
If r is odd,  row is increasing.
If r is even, row is decreasing.

Mathematica

T[n_, k_] := ((n+k)^2 - 2(n+k) + 4 - (n+3k-2)(-1)^(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 05 2019 *)

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

Crossrefs

Cf. A211394, A002260, A004736, A003057.

Sequence in context: A112282 A098866 A144689 * A199180 A197265 A198107

Adjacent sequences: A221212 A221213 A221214 * A221216 A221217 A221218

Keyword

nonn,tabl

Author

Boris Putievskiy, Feb 22 2013

Status

approved