A221217 - OEIS

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A221217

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

1

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

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

. . .

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

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

. . .

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 upwards.

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

As linear sequence

a(n) = (A003057(n)^2-2*A002260(n)+3-A002024(n)*(1+2*(-1)^A003056(n)))/2;

a(n) = ((t+2)^2-2*i+3-(t+1)*(1+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...4..15..11..28..22...

5....3..14..10..27..21..44...

2...13...9..26..20..43..35...

12...8..25..19..42..34..63...

7...24..18..41..33..62..52...

23..17..40..32..61..51..86...

16..39..31..60..50..85..73...

. . .

The start of the sequence as triangle array read by rows:

1;

6,5;

4,3,2;

15,14,13,12;

11,10,9,8,7;

28,27,26,25,24,23;

22,21,20,19,18,17,16;

. . .

Row number r consecutive contains r numbers in decreasing order.

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

CROSSREFS

Cf. A211394, A221215, A002260, A004736, A003057, A002024;

table T(n,k) contains: in rows A084849, A000384, A014106, A014105, A014107, A091823, A071355, A091823, A071355, A100040, A130861, A100041;

in columns A130883, A096376, A033816, A100037, A100038, A100039;

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

Sequence in context: A055117 A090861 A132670 * A018868 A302712 A125089

Adjacent sequences: A221214 A221215 A221216 * A221218 A221219 A221220

KEYWORD

nonn,tabl

AUTHOR

Boris Putievskiy, Feb 22 2013

STATUS

approved