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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified September 26 14:21 EDT 2022. Contains 356999 sequences. (Running on oeis4.)