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A209293 Inverse permutation of A185180. 5
1, 2, 3, 5, 4, 6, 8, 9, 7, 10, 13, 12, 14, 11, 15, 18, 19, 17, 20, 16, 21, 25, 24, 26, 23, 27, 22, 28, 32, 33, 31, 34, 30, 35, 29, 36, 41, 40, 42, 39, 43, 38, 44, 37, 45, 50, 51, 49, 52, 48, 53, 47, 54, 46, 55, 61, 60, 62, 59, 63, 58, 64, 57, 65, 56, 66, 72, 73, 71, 74, 70, 75, 69, 76, 68, 77, 67 (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) by diagonals. The order of the list

if n is odd  - T(n-1,2),T(n-3,4),...,T(2,n-1),T(1,n),T(3,n-2),...T(n,1).

if n is even - T(n-1,2),T(n-3,4),...,T(3,n-2),T(1,n),T(2,n-1),...T(n,1).

Table T(n,k) contains:

Column number 1 A000217,

column number 2 A000124,

column number 3 A000096,

column number 4 A152948,

column number 5 A034856,

column number 6 A152950,

column number 7 A055998.

Row    numder 1 A000982,

row    number 2 A097063.

LINKS

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO]

Eric W. Weisstein, MathWorld: Pairing functions

Index entries for sequences that are permutations of the natural numbers

FORMULA

As table T(n,k) read by antidiagonals

T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.

As linear sequence

a(n) = (m1+m2-1)*(m1+m2-2)/2+m1, where

m1 = int((i+j)/2)+int(i/2)*(-1)^(i+t+1),

m2 = int((i+j+1)/2)+int(i/2)*(-1)^(i+t),

t = int((math.sqrt(8*n-7) - 1)/ 2),

i = n-t*(t+1)/2,

j = (t*t+3*t+4)/2-n.

EXAMPLE

The start of the sequence as table:

1....2...5...8..13..18...25...32...41...

3....4...9..12..19..24...33...40...51...

6....7..14..17..26..31...42...49...62...

10..11..20..23..34..39...52...59...74...

15..16..27..30..43..48...63...70...87...

21..22..35..38..53..58...75...82..101...

28..29..44..47..64..69...88...95..116...

36..37..54..57..76..81..102..109..132...

45..46..65..68..89..94..117..124..149...

. . .

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

1;

2,3;

5,4,6;

8,9,7,10;

13,12,14,11,15;

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

25,24,26,23,27,22,28;

32,33,31,34,30,35,29,36;

41,40,42,39,43,38,44,37,45;

. . .

Row number r contains permutation from r numbers:

if r is odd  ceiling(r^2/2), ceiling(r^2/2)+1, ceiling(r^2/2)-1, ceiling(r^2/2)+2, ceiling(r^2/2)-2,...r*(r+1)/2;

if r is even ceiling(r^2/2), ceiling(r^2/2)-1, ceiling(r^2/2)+1, ceiling(r^2/2)-2, ceiling(r^2/2)+2,...r*(r+1)/2;

MATHEMATICA

max = 10; row[n_] := Table[Ceiling[(n + k - 1)^2/2] + If[OddQ[k], 1, -1]*Floor[n/2], {k, 1, max}]; t = Table[row[n], {n, 1, max}]; Table[t[[n - k + 1, k]], {n, 1, max}, {k, n, 1, -1}] // Flatten (* Jean-Fran├žois Alcover, Jan 17 2013 *)

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

m1=int((i+j)/2)+int(i/2)*(-1)**(i+t+1)

m2=int((i+j+1)/2)+int(i/2)*(-1)**(i+t)

m=(m1+m2-1)*(m1+m2-2)/2+m1

CROSSREFS

Cf. A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

Sequence in context: A073291 A073280 A073884 * A328021 A097290 A279344

Adjacent sequences:  A209290 A209291 A209292 * A209294 A209295 A209296

KEYWORD

nonn,tabl

AUTHOR

Boris Putievskiy, Jan 16 2013

STATUS

approved

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Last modified October 17 04:09 EDT 2019. Contains 328106 sequences. (Running on oeis4.)