A064790 -id:A064790 - cp's OEIS frontend

A060734 Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are read by antidiagonals downwards.

1, 4, 2, 9, 3, 5, 16, 8, 6, 10, 25, 15, 7, 11, 17, 36, 24, 14, 12, 18, 26, 49, 35, 23, 13, 19, 27, 37, 64, 48, 34, 22, 20, 28, 38, 50, 81, 63, 47, 33, 21, 29, 39, 51, 65, 100, 80, 62, 46, 32, 30, 40, 52, 66, 82, 121, 99, 79, 61, 45, 31, 41, 53, 67, 83, 101

Offset: 1

Frank Ellermann, Apr 23 2001

A simple permutation of natural numbers.
Parity of the sequence is given by A057211 (n-th run has length n). - Jeremy Gardiner, Dec 26 2008
The square with corners T(1,1)=1 and T(n,n)=n^2-n+1 is occupied by the numbers 1,2,...,n^2. - Clark Kimberling, Feb 01 2011
a(n) is pairing function - function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} - the set of integer positive numbers. - Boris Putievskiy, Dec 17 2012

			Northwest corner:
.1  4  9 16 ..  => a(1) =  1
.2  3  8 15 ..  => a(2) =  4, a(3) = 2
.5  6  7 14 ..  => a(4) =  9, a(5) = 3, a(6) = 5
10 11 12 13 ..  => a(7) = 16, a(8) = 8, a(9) = 6, a(10)=10

Alois P. Heinz, Rows n = 1..141 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060736. Inverse: A064790.

Cf. A185725, A185726, A185728.

T:= (n,k)-> `if`(n<=k, k^2-n+1, (n-1)^2+k):
seq(seq(T(n, d-n), n=1..d-1), d=2..15);

f[n_, k_]:=k^2-n+1/; k>=n;
f[n_, k_]:=(n-1)^2+k/; kClark Kimberling, Feb 01 2011 *)

T(n,k) = (n-1)^2+k, T(k, n)=n^2+1-k, 1 <= k <= n.
From Clark Kimberling, Feb 01 2011: (Start)
T(1,k) = k^2 (A000290).
T(n,n) = n^2-n+1 (A002061).
T(n,1) = (n-1)^2+1 (A002522). (End)

Corrected by Jeremy Gardiner, Dec 26 2008

A188568 Enumeration table T(n,k) by descending antidiagonals. The order of the list - if n is odd: T(n,1), T(2,n-1), T(n-2,3), ..., T(n-1,2), T(1,n); if n is even: T(1,n), T(n-1,2), T(3,n-2), ..., T(2,n-1), T(n,1).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Dec 27 2012

Self-inverse 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.
Call a "layer" a pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). This table read layer by layer clockwise is A194280. This table read by boustrophedonic ("ox-plowing") method - layer clockwise, layer counterclockwise and so on - is A064790. - Boris Putievskiy, Mar 14 2013

			The start of the sequence as table:
   1,  2,  6,  7, 15, 16, 28, ...
   3,  5,  9, 12, 20, 23, 35, ...
   4,  8, 13, 18, 26, 31, 43, ...
  10, 14, 19, 25, 33, 40, 52, ...
  11, 17, 24, 32, 41, 50, 62, ...
  21, 27, 34, 42, 51, 61, 73, ...
  22, 30, 39, 49, 60, 72, 85, ...
  ...
The start of the sequence as triangular array read by rows:
   1;
   2,  3;
   6,  5,  4;
   7,  9,  8, 10;
  15, 12, 13, 14, 11;
  16, 20, 18, 19, 17, 21;
  28, 23, 26, 25, 24, 27, 22;
  ...
Row number k contains permutation of the k numbers:
{ (k^2-k+2)/2, (k^2-k+2)/2 + 1, ..., (k^2+k-2)/2 + 1 }.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A056011, A056023, A057027, A064578, A194981, A194982, inverse functions A208233, A208234, A194280, A064790.

a[n_] := Module[{t, i, j},
t = Floor[(Sqrt[8n-7]-1)/2];
i = n-t(t+1)/2;
j = (t^2+3t+4)/2-n;
((i+j-1)(i+j-2) + ((-1)^Max[i,j]+1)i - ((-1)^Max[i,j]-1)j)/2];
Array[a, 55] (* Jean-François Alcover, Jan 26 2019 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
m=((i+j-1)*(i+j-2)+((-1)**max(i,j)+1)*i-((-1)**max(i,j)-1)*j)/2

a(n) = ((i+j-1)*(i+j-2)+((-1)^max(i,j)+1)*i-((-1)^max(i,j)-1)*j)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2].

A194280 Inverse permutation to A081344.

Original entry on oeis.org

1, 2, 5, 3, 6, 9, 13, 8, 4, 7, 12, 18, 25, 19, 14, 10, 15, 20, 26, 33, 41, 32, 24, 17, 11, 16, 23, 31, 40, 50, 61, 51, 42, 34, 27, 21, 28, 35, 43, 52, 62, 73, 85, 72, 60, 49, 39, 30, 22, 29, 38, 48, 59, 71, 84, 98, 113

Offset: 1

Boris Putievskiy, Dec 23 2012

nonn

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.
Call a "layer" a pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). This sequence is A188568 as table read layer by layer clockwise.
The same table A188568 read  by boustrophedon ("ox-plowing") method - layer clockwise, layer counterclockwise and so on - is A064790. - Boris Putievskiy, Mar 14 2013

			From _Boris Putievskiy_, Mar 14 2013: (Start)
The start of the sequence as table:
  1....2...6...7..15..16..28...
  3....5...9..12..20..23..35...
  4....8..13..18..26..31..43...
  10..14..19..25..33..40..52...
  11..17..24..32..41..50..62...
  21..27..34..42..51..61..73...
  22..30..39..49..60..72..85...
  ...
The start of the sequence as triangular array read by rows:
  1;
  2,5,3;
  6,9,13,8,4;
  7,12,18,25,19,14,10;
  15,20,26,33,41,32,24,17,11;
  16,23,31,40,50,61,51,42,34,27,21;
  28,35,43,52,62,73,85,72,60,49,39,30,22;
  ...
Row number r contains 2*r-1 numbers. (End)

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Index entries for sequences that are permutations of the natural numbers

Cf. A081344, A064790, A188568.

t=int(math.sqrt(n-1))+1
i=(t % 2)*min(t,n-(t-1)**2) + ((t+1) % 2)*min(t,t**2-n+1)
j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)
m=(i+j-1)*(i+j-2)/2+j

a(n) = (i+j-1)*(i+j-2)/2+j, where
i = mod(t;2)*min{t; n - (t - 1)^2} + mod(t + 1; 2)*min{t; t^2 - n + 1}
j = mod(t;2)*min{t; t^2 - n + 1} + mod(t + 1; 2)*min{t; n - (t - 1)^2},
t = int(math.sqrt(n-1))+1.

A064788 Inverse permutation to A060736.

Original entry on oeis.org

1, 2, 5, 3, 4, 8, 13, 9, 6, 7, 12, 18, 25, 19, 14, 10, 11, 17, 24, 32, 41, 33, 26, 20, 15, 16, 23, 31, 40, 50, 61, 51, 42, 34, 27, 21, 22, 30, 39, 49, 60, 72, 85, 73, 62, 52, 43, 35, 28, 29, 38, 48, 59, 71, 84, 98, 113, 99, 86, 74, 63, 53, 44, 36, 37, 47, 58, 70, 83, 97, 112

Offset: 1

N. J. A. Sloane, Oct 20 2001

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's MathWorld, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A064790, A194280.

A064788(n)={my(t=sqrtint(n-1)+1); A000217(min(t,t^2-n+1)-2+n=min(t, n-(t-1)^2))+n} \\ recall: A000217(n)=(n+1)*n/2 \\ M. F. Hasler, Jan 13 2013

a(n) = (i+j-1)*(i+j-2)/2+j, where i=min(t; t^2-n+1), j=min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1. - Boris Putievskiy, Dec 24 2012

More terms from David Wasserman, Jan 15 2002

cp's OEIS Frontend

A060734 Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are read by antidiagonals downwards.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A188568 Enumeration table T(n,k) by descending antidiagonals. The order of the list - if n is odd: T(n,1), T(2,n-1), T(n-2,3), ..., T(n-1,2), T(1,n); if n is even: T(1,n), T(n-1,2), T(3,n-2), ..., T(2,n-1), T(n,1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A194280 Inverse permutation to A081344.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A064788 Inverse permutation to A060736.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions