A194280 -id:A194280 - cp's OEIS frontend

A081344 Natural numbers in square maze arrangement, read by antidiagonals.

1, 2, 4, 9, 3, 5, 10, 8, 6, 16, 25, 11, 7, 15, 17, 26, 24, 12, 14, 18, 36, 49, 27, 23, 13, 19, 35, 37, 50, 48, 28, 22, 20, 34, 38, 64, 81, 51, 47, 29, 21, 33, 39, 63, 65, 82, 80, 52, 46, 30, 32, 40, 62, 66, 100, 121, 83, 79, 53, 45, 31, 41, 61, 67, 99, 101, 122, 120, 84, 78, 54

Offset: 1

Paul Barry, Mar 19 2003

Arrange the natural numbers by taking clockwise and counterclockwise turns. Begin (LL) and then repeat (RRR)(LLL).
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. - Boris Putievskiy, Dec 16 2012
For generalizations see A219159, A213928. - Boris Putievskiy, Mar 10 2013

			The start of the sequence as table T(i,j), i,j > 0:
   1   4    5    16 ...
   2   3    6    15 ...
   9   8    7    14 ...
  10  11   12    13 ...
  ....

Boris Putievskiy, Rows n = 1..100 of triangle, flattened
Boris Putievskiy, Transformations 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

Cf. A219159, A213928. The main diagonal is A002061. The following appear within interlaced sequences: A016754, A001844, A053755, A004120. The first row is A081345. The first column is A081346. The inverse permutation A194280, the first inverse function (numbers of rows) A220603, the second inverse function (numbers of columns) A220604.

T[n_, k_] := T[n, k] = Which[OddQ[n] && k==1, n^2, EvenQ[k] && n==1, k^2, EvenQ[n] && k==1, T[n-1, 1]+1, OddQ[k] && n==1, T[1, k-1]+1, k <= n, T[n, k-1]+1 - 2 Mod[n, 2], True, T[n-1, k]-1 + 2 Mod[k, 2]]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 20 2019 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if j >= i:
     m=(j-1)**2 + j + (j-i)*(-1)**(j-1)
else:
     m=(i-1)**2 + i - (i-j)*(-1)**(i-1)
# Boris Putievskiy, Dec 19 2012

from math import isqrt
def A081344(n):
    t = (k:=isqrt(m:=n<<1))+((m<<2)>(k<<2)*(k+1)+1)-1
    i, j = n-(t*(t+1)>>1), (t*(t+3)>>1)+2-n
    r = max(i,j)
    return (r-1)**2+r+(j-i if r&1 else i-j) # Chai Wah Wu, Nov 04 2024

From Boris Putievskiy, Dec 19 2012: (Start)
a(n) = (i-1)^2 + i + (i-j)*(-1)^(i-1) if i >= j,
a(n) = (j-1)^2 + j - (j-i)*(-1)^(j-1) if i <  j,
where
i = n - t*(t+1)/2,
j = (t*t + 3*t + 4)/2-n,
t = floor((-1 + sqrt(8*n-7))/2). (End)
Enumeration by boustrophedonic ("ox-plowing") method: If i >= j: T(i,j)=(i-1)^2+i + (i-j)*(-1)^(i-1), if i  < j: T(i,j)=(j-1)^2+j - (j-i)*(-1)^(j-1). - Boris Putievskiy, Dec 19 2012
T(i,j) = m^2 - m + 1 - (i - j)*(-1)^m where m = max(i,j). - Ziad Ahmed, Jun 09 2025

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].

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

A064790 Inverse permutation to A060734.

Original entry on oeis.org

1, 3, 5, 2, 6, 9, 13, 8, 4, 10, 14, 19, 25, 18, 12, 7, 15, 20, 26, 33, 41, 32, 24, 17, 11, 21, 27, 34, 42, 51, 61, 50, 40, 31, 23, 16, 28, 35, 43, 52, 62, 73, 85, 72, 60, 49, 39, 30, 22, 36, 44, 53, 63, 74, 86, 99, 113, 98, 84, 71, 59, 48, 38, 29, 45, 54, 64, 75, 87, 100, 114

Offset: 1

N. J. A. Sloane, Oct 20 2001

From Boris Putievskiy, Mar 14 2013: (Start)
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.
Layer is 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  by boustrophedonic ("ox-plowing") method - layer clockwise, layer counterclockwise and so. The same table A188568 read layer by layer clockwise is A194280. (End)

			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;
  3,5,2;
  6,9,13,8,4;
  10,14,19,25,18,12,7;
  15,20,26,33,41,32,24,17,11;
  21,27,34,42,51,61,50,40,31,23,16;
  28,35,43,52,62,73,85,72,60,49,39,30,22;
  ...
Row number r contains 2*r-1 numbers. (End)

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

Cf. A060734, A064788, A188568, A194280.

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

A219159 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 2 layers clockwise and so on. T(n,k) read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Feb 19 2013

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.
In general, let m be natural number. Layer is pair of sides of square table T(n,k) from T(1,n) to T(n,n) and  from T(n,n) to T(n,1). Natural numbers placed in the table T(n,k) layer by layer. The order of placement - at the beginning m layers counterclockwise, next m layers clockwise and so on. T(n,k) read by antidiagonals.
For m = 1 the result is A081344. This sequence is result for m = 2.

			The start of the sequence as table.
The direction of the placement denotes by ">" and  "v".
            v..v           v...v
  .>1...4...5..10..25..36..37..50...
  .>2...3...6..11..24..35..38..51...
  ..9...8...7..12..23..34..39..52...
  .16..15..14..13..22..33..40..53...
  >17..18..19..20..21..32..41..54...
  >26..27..28..29..30..31..42..55...
  .49..48..47..46..45..44..43..56...
  .64..63..62..61..60..59..58..57...
  . . .
The start of the sequence as triangle array read by rows:
   1;
   4,  2;
   5,  3,  9;
  10,  6,  8, 16;
  25, 11,  7, 15, 17;
  36, 24, 12, 14, 18, 26;
  37, 35, 23, 13, 19, 27, 49;
  50, 38, 34, 22, 20, 28, 48, 64;
   ...

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

Cf. A081344, A194280.

T[n_, k_] := If[k >= n, ((1 + (-1)^Floor[(k-1)/2])(k^2 - n + 1) - (-1 + (-1)^Floor[(k-1)/2])((k-1)^2 + n))/2, ((1 + (-1)^(Floor[(n-1)/2] + 1))(n^2 - k + 1) - (-1 + (-1)^(Floor[(n-1)/2] + 1))((n-1)^2 + k))/2];
Table[T[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 11 2018 *)

T(n,k) := if  k >= n then ((1 + (-1)^floor((k - 1)/2))*(k^2 - n + 1) - (-1 + (-1)^floor((k - 1)/2))*((k - 1)^2 + n))/2 else ((1 + (-1)^(floor((n - 1)/2) + 1))*(n^2 - k + 1) - (-1 + (-1)^(floor((n - 1)/2) + 1))*((n - 1)^2 +k))/2$
create_list(T(k, n - k), n, 1, 12, k, 1, n - 1); /* Franck Maminirina Ramaharo, Dec 11 2018 */

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if j >= i:
   result=((1+(-1)**int((j-1)/2))*(j**2-i+1)-(-1+(-1)**int((j-1)/2))*((j-1)**2 +i))/2
else:
   result=((1+(-1)**(int((i-1)/2)+1))*(i**2-j+1)-(-1+(-1)**(int((i-1)/2)+1))*((i-1)**2 +j))/2

For general case.
As table
T(n,k) = ((1 + (-1)^floor((k - 1)/m))*(k^2 - n + 1) - (-1 + (-1)^floor((k - 1)/m))*((k - 1)^2 + n))/2, if  k >= n; T(n,k) = ((1 + (-1)^(floor((n - 1)/m) + 1))*(n^2 - k + 1) - (-1 + (-1)^(floor((n - 1)/m) + 1))*((n - 1)^2 +k))/2, if  n > k.
As linear sequence
a(n) = ((1 + (-1)^floor((j - 1)/m))*(j^2 - i + 1) - (-1 + (-1)^floor((j - 1)/m))*((j - 1)^2 + i))/2, if  j >= i; a(n) = ((1 + (-1)^(floor((i - 1)/m) + 1))*(i^2 - j + 1) - (-1 + (-1)^(floor((i - 1)/m) + 1))*((i - 1)^2 + j))/2, if  i > j; where i = n - t*(t + 1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2).
For this sequence.
As table
T(n,k) = ((1 + (-1)^floor((k - 1)/2))*(k^2 - n + 1) - (-1 + (-1)^floor((k - 1)/2))*((k - 1)^2 + n))/2, if  k >= n; T(n,k) = ((1 + (-1)^(floor((n - 1)/2) + 1))*(n^2 - k + 1) - (-1 + (-1)^(floor((n - 1)/2) + 1))*((n - 1)^2 + k))/2, if  n > k.
As linear sequence
a(n) = ((1 + (-1)^floor((j - 1)/2))*(j^2 - i + 1) - (-1 + (-1)^floor((j - 1)/2))*((j - 1)^2 + i))/2, if  j >= i; a(n) = ((1 + (-1)^(floor((i - 1)/2) + 1))*(i^2 - j + 1) - (-1 + (-1)^(floor((i - 1)/2) + 1))*((i - 1)^2 + j))/2, if  i > j; where i = n - t*(t + 1)/2, j = (t*t +3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2).

A213928 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Mar 06 2013

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.In general, let b(z) be a sequence of integer numbers. Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Natural numbers placed in table T(n,k) layer by layer. The order of placement - layer is counterclockwise, if b(z) is odd; layer is clockwise if b(z) is even. T(n,k) read by antidiagonals.For A219159 - the order of the placement - at the beginning m layers counterclockwise, next m layers clockwise and so on - b(z)=floor((z-1)/m)+1. For this sequence b(z)=z^2 mod 3.

			The start of the sequence as table.
The direction of the placement denotes by ">" and  "v".
  ..........v...........v...........v
  >1....4...5..16..25..26..49..64..65...
  >2....3...6..15..24..27..48..63..66...
  .9....8...7..14..23..28..47..62..67...
  >10..11..12..13..22..29..46..61..68...
  >17..18..19..20..21..30..45..60..69...
  .36..35..34..33..32..31..44..59..70...
  >37..38..39..40..41..42..43..58..71...
  >50..51..52..53..54..55..56..57..72...
  .81..80..79..78..77..76..75..74..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  4,2;
  5,3,9;
  16,6,8,10;
  25,15,7,11,17;
  26,24,14,12,18,36;
  49,27,23,13,19,35,37;
  64,48,28,22,20,34,38,50;
  65,63,47,29,21,33,39,51,81;
  . . .

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

Cf. A219159, A081344, A194280, A042964, A130196, A011655, A220516.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if j>=i:
   result=((1+(-1)**(j**2%3-1))*(j**2-i+1)-(-1+(-1)**(j**2%3-1))*((j-1)**2 +i))/2
else:
   result=((1+(-1)**(i**2%3))*(i**2-j+1)-(-1+(-1)**(i**2%3))*((i-1)**2 +j))/2

For general case.
As table
T(n,k) = ((1+(-1)^(b(k)-1))*(k^2-n+1)-(-1+(-1)^(b(k)-1))*((k-1)^2 +n))/2, if  k >= n;
T(n,k) = ((1+(-1)^b(n))*(n^2-k+1)-(-1+(-1)^b(n))*((n-1)^2 +k))/2, if  n >k.
As linear sequence
a(n) = ((1+(-1)^(b(j)-1))*(j^2-i+1)-(-1+(-1)^(b(j)-1))*((j-1)^2 +i))/2, if  j >= i;
a(n) = ((1+(-1)^b(i))*(i^2-j+1)-(-1+(-1)^b(i))*((i-1)^2 +j))/2, if  i >j;
where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
For this sequence b(z)=z^2 mod 3.
As table
T(n,k) = ((1+(-1)^(k^2 mod 3-1))*(k^2-n+1)-(-1+(-1)^(k^2 mod 3-1))*((k-1)^2 +n))/2, if  k >= n;
T(n,k) = ((1+(-1)^(n^2 mod 3))*(n^2-k+1)-(-1+(-1)^(n^2 mod 3))*((n-1)^2 +k))/2, if  n >k.
As linear sequence
a(n) = ((1+(-1)^(j^2 mod 3-1))*(j^2-i+1)-(-1+(-1)^(j^2 mod 3-1))*((j-1)^2 +i))/2, if  j >= i;
a(n) = ((1+(-1)^(i^2 mod 3))*(i^2-j+1)-(-1+(-1)^(i^2 mod 3))*((i-1)^2 +j))/2, if  i >j;
where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

cp's OEIS Frontend

A081344 Natural numbers in square maze arrangement, read by antidiagonals.

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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).

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A064788 Inverse permutation to A060736.

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A064790 Inverse permutation to A060734.

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A219159 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 2 layers clockwise and so on. T(n,k) read by antidiagonals.

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A213928 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.

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