A060736 -id:A060736 - cp's OEIS frontend

A064788 Inverse permutation to A060736.

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

A061349 Sum of antidiagonals of A060736.

Original entry on oeis.org

0, 1, 6, 17, 40, 75, 130, 203, 304, 429, 590, 781, 1016, 1287, 1610, 1975, 2400, 2873, 3414, 4009, 4680, 5411, 6226, 7107, 8080, 9125, 10270, 11493, 12824, 14239, 15770, 17391, 19136, 20977, 22950, 25025, 27240, 29563, 32034, 34619, 37360, 40221

Offset: 0

Henry Bottomley, Jun 07 2001

a(1) = 1, a(2) = 2+4=6, a(3) = 5+3+9=17, a(4) = 10+6+8+16=40.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A001844, A002378, A004526, A005900, A006918, A060736.

LinearRecurrence[{2,1,-4,1,2,-1},{0,1,6,17,40,75},50] (* Harvey P. Dale, Oct 17 2021 *)
Accumulate[Table[n^2 + (n - 1)^2 - Floor[((n-1)/2)]*Floor[((n+1)/2)],{n,41}]] (* Stefano Spezia, Jun 05 2023 *)

concat(0, Vec(x*(x^4+4*x^3+4*x^2+4*x+1)/((x-1)^4*(x+1)^2) + O(x^100))) \\ Colin Barker, Sep 13 2014

a(n) = A005900(n) - A006918(n).
a(n) = a(n-1) + A001844(n-1) - A002378(A004526(n-1)).
a(n) = a(n-1) + n^2 + (n - 1)^2 - floor((n-1)/2)*floor((n+1)/2).
If n is odd then a(n) = (7*n^3 + 5*n)/12;
If n is even then a(n) = (7*n^3 + 8*n)/12.
From Colin Barker, Sep 13 2014: (Start)
a(n) = (n*(13 + 3*(-1)^n + 14*n^2))/24.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: x*(x^4 + 4*x^3 + 4*x^2 + 4*x + 1)/((x - 1)^4*(x + 1)^2). (End)
E.g.f.: x*((12 + 21*x + 7*x^2)*cosh(x) + (15 + 21*x + 7*x^2)*sinh(x))/12. - Stefano Spezia, Jun 05 2023

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.

Original entry on oeis.org

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

A220508 T(n,k) = n^2 + k if k <= n, otherwise T(n,k) = k*(k + 2) - n; square array T(n,k) read by ascending antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

0, 1, 3, 4, 2, 8, 9, 5, 7, 15, 16, 10, 6, 14, 24, 25, 17, 11, 13, 23, 35, 36, 26, 18, 12, 22, 34, 48, 49, 37, 27, 19, 21, 33, 47, 63, 64, 50, 38, 28, 20, 32, 46, 62, 80, 81, 65, 51, 39, 29, 31, 45, 61, 79, 99, 100, 82, 66, 52, 40, 30, 44, 60, 78, 98, 120

Offset: 0

Omar E. Pol, Feb 09 2013

This sequence consists of 0 together with a permutation of the natural numbers. The nonnegative integers (A001477) are arranged in the successive layers from T(0,0) = 0. The n-th layer start with T(n,1) = n^2. The n-th layer is formed by the first n+1 elements of row n and the first n elements in increasing order of the column n.
The first antidiagonal is formed by odd numbers: 1, 3. The second antidiagonal is formed by even numbers: 4, 2, 8. The third antidiagonal is formed by odd numbers: 9, 5, 7, 15. And so on.
It appears that in the n-th layer there is at least a prime number <= g and also there is at least a prime number > g, where g is the number on the main diagonal, the n-th oblong number A002378(n), if n >= 1.

			The second layer is [4, 5, 6, 7, 8] which looks like this:
  .  .  8
  .  .  7,
  4, 5, 6,
Square array T(0,0)..T(10,10) begins:
    0,   3,   8,  15,  24,  35,  48,  63,  80,  99, 120,...
    1,   2,   7,  14,  23,  34,  47,  62,  79,  98, 119,...
    4,   5,   6,  13,  22,  33,  46,  61,  78,  97, 118,...
    9,  10,  11,  12,  21,  32,  45,  60,  77,  96, 117,...
   16,  17,  18,  19,  20,  31,  44,  59,  76,  95, 118,...
   25,  26,  27,  28,  29,  30,  43,  58,  75,  94, 117,...
   36,  37,  38,  39,  40,  41,  42,  57,  74,  93, 114,...
   49,  50,  51,  52,  53,  54,  55,  56,  73,  92, 113,...
   64,  65,  66,  67,  68,  69,  70,  71,  72,  91, 112,...
   81,  82,  83,  84,  85,  86,  87,  88,  89,  90, 111,...
  100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110,...
  ...

Index entries for sequences that are permutations of the natural numbers

Column 1 is A000290. Main diagonal is A002378. Column 2 is essentially A002522. Row 1 is A005563. Row 2 gives the absolute terms of A008865.

Cf. A000005, A000040, A002061, A002620, A014206, A014209, A027688, A028387, A028552, A060736, A220516.

From Petros Hadjicostas, Mar 10 2021: (Start)
T(n,k) = (A342354(n,k) - 1)/2.
O.g.f.: (x^4*y^3 + 3*x^3*y^4 + x^4*y^2 - 10*x^3*y^3 - x^2*y^4 + 3*x^3*y^2 + x^2*y^3 - 4*x^3*y + 8*x^2*y^2 + 3*x^2*y + x*y^2 + x^2 - 10*x*y - y^2 + x + 3*y)/((1 - x)^3*(1 - y)^3*(1 - x*y)^2). (End)

Name edited by Petros Hadjicostas, Mar 10 2021

A213921 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places clockwise. Table T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 8, 9, 13, 17, 14, 6, 16, 21, 26, 22, 11, 12, 25, 31, 37, 32, 18, 15, 20, 36, 43, 50, 44, 27, 23, 24, 30, 49, 57, 65, 58, 38, 33, 19, 35, 42, 64, 73, 82, 74, 51, 45, 28, 29, 48, 56, 81, 91, 101, 92, 66, 59, 39, 34, 41, 63, 72, 100, 111

Offset: 1

Boris Putievskiy, Mar 05 2013

A 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.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
T(1,1)=1;
T(1,2), T(2,1), T(2,2);
. . .
T(1,n), T(3,n), ... T(n,3), T(n,1), T(2,n), T(4,n), ... T(n,4), T(n,2);
...

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   8  14  22  32 ...
   7   9   6  11  18  27 ...
  13  16  12  15  23  33 ...
  21  25  20  24  19  28 ...
  31  36  30  35  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  8,  9, 13;
  17, 14,  6, 16, 21;
  26, 22, 11, 12, 25, 31;
  ...

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 W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A002522, A014206, A059100, A027688, A117950, A027689, A087475, A027690, A117951, A027691, A114949, A027692, A117619; in columns A002061, A000290, A002378, A005563, A028387, A008865, A028552, A028872, A014209, A028347, A028875.

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

As a table:
T(n,k) = n*n - (k mod 2)*n + 2 - floor((k+2)/2), if n>k;
T(n,k) = k*k - ((n mod 2)+1)*k + floor((n+3)/2), if n<=k.
As a linear sequence:
a(n) = i*i - (j mod 2)*i + 2 - floor((j+2)/2), if i>j;
a(n) = j*j - ((i mod 2)+1)*j + floor((i+3)/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).

A213922 Natural numbers placed in table T(n,k) layer by layer. The order of placement: T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1). Table T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 3, 4, 8, 2, 9, 15, 6, 7, 16, 24, 13, 5, 14, 25, 35, 22, 11, 12, 23, 36, 48, 33, 20, 10, 21, 34, 49, 63, 46, 31, 18, 19, 32, 47, 64, 80, 61, 44, 29, 17, 30, 45, 62, 81, 99, 78, 59, 42, 27, 28, 43, 60, 79, 100, 120, 97, 76, 57, 40, 26, 41, 58, 77, 98, 121

Offset: 1

Boris Putievskiy, Mar 05 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.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
T(1,1)=1;
T(2,2), T(1,2), T(2,1);
...
T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1);
...

			The start of the sequence as a table:
   1,  3,  8, 15, 24, 35, ...
   4,  2,  6, 13, 22, 33, ...
   9,  7,  5, 11, 20, 31, ...
  16, 14, 12, 10, 18, 29, ...
  25, 23, 21, 19, 17, 27, ...
  36, 34, 32, 30, 28, 26, ...
...
The start of the sequence as triangular array read by rows:
   1;
   3,  4;
   8,  2,  9;
  15,  6,  7, 16;
  24, 13,  5, 14, 25;
  35, 22, 11, 12, 23, 36;
  ...

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 W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A005563, A028872, A028875, A028881, A028560, A116711; in columns A000290, A008865, A028347, A028878, A028884.

f[n_, k_] := n^2 - 2*k + 2 /; n >= k; f[n_, k_] := k^2 - 2*n + 1 /; n < k; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 10}]]; Table[f[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Aug 19 2017 *)

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

As a table,
  T(n,k) = n*n - 2*k + 2, if n >= k;
  T(n,k) = k*k - 2*n + 1, if n < k.
As a linear sequence,
  a(n) = i*i - 2*j + 2, if i >= j;
  a(n) = j*j - 2*i + 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).

A194195 First inverse function (numbers of rows) for pairing function A060734.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 3, 2, 1, 4, 4, 4, 4, 3, 2, 1, 5, 5, 5, 5, 5, 4, 3, 2, 1, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 1

Boris Putievskiy, Dec 21 2012

The sequence is the second inverse function (numbers of columns) for pairing function A060736.

			The start of the sequence as triangle array read by rows:
1;
2,2,1;
3,3,3,2,1;
4,4,4,4,3,2,1;
. . .
Row number k contains 2k-1 numbers k,k,...k,k-1,k-2,...1 (k times repetition "k").

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A060734, A060736, A220603, A220604

f[n_]:=Module[{t=Floor[Sqrt[n-1]]+1},Min[t,t^2-n+1]]; Array[f,80] (* Harvey P. Dale, Dec 31 2012 *)

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

a(n) = min{t; t^2 - n + 1}, where t=floor(sqrt(n-1))+1.

A194258 Second inverse function (numbers of columns) for pairing function A060734.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 3, 3, 1, 2, 3, 4, 4, 4, 4, 1, 2, 3, 4, 5, 5, 5, 5, 5, 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 1, 2, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 1, 2, 3, 4, 5, 6

Offset: 1

Boris Putievskiy, Dec 21 2012

The sequence is the first inverse function (numbers of rows) for pairing function A060736.

			The start of the sequence as triangle array read by rows:
1;
1,2,2;
1,2,3,3,3;
1,2,3,4,4,4,4;
. . .
Row number k contains 2k-1 numbers 1,2,...k-1,k,k,...k (k times repetition "k").

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A060734, A060736, A220603, A220604.

Flatten[Table[Join[Range[n-1],Table[n,{n}]],{n,10}]] (* Harvey P. Dale, Jun 23 2013 *)

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

a(n) = min{t; n - (t - 1)^2}, where t=floor(sqrt(n-1))+1.

A214928 A209293 as table read layer by layer clockwise.

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 14, 7, 6, 8, 12, 17, 23, 20, 11, 10, 13, 19, 26, 34, 43, 30, 27, 16, 15, 18, 24, 31, 39, 48, 58, 53, 38, 35, 22, 21, 25, 33, 42, 52, 63, 75, 88, 69, 64, 47, 44, 29, 28, 32, 40, 49, 59, 70, 82, 95, 109, 102, 81, 76, 57, 54, 37, 36, 41, 51, 62

Offset: 1

Boris Putievskiy, Mar 11 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.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). The order of the list:
T(1,1)=1;
T(1,2), T(2,2), T(2,1);
. . .
T(1,n), T(2,n), ... T(n-1,n), T(n,n), T(n,n-1), ... T(n,1);
. . .

			The start of the sequence as table:
  1....2...5...8..13..18...
  3....4...9..12..19..24...
  6....7..14..17..26..31...
  10..11..20..23..34..39...
  15..16..27..30..43..48...
  21..22..35..38..53..58...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  2,4,3;
  5,9,14,7,6;
  8,12,17,23,20,11,10;
  13,19,26,34,43,30,27,16,15;
  18,24,31,39,48,58,53,38,35,22,21;
  . . .
Row number r contains 2*r-1 numbers.

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. A209293, A209279, A209278, A185180, A060734, A060736; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

t=int((math.sqrt(n-1)))+1
i=min(t,n-(t-1)**2)
j=min(t,t**2-n+1)
m1=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
result=(m1+m2-1)*(m1+m2-2)/2+m1

As table
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=floor((i+j)/2) + floor(i/2)*(-1)^(2*i+j-1), m2=int((i+j+1)/2)+int(i/2)*(-1)^(2*i+j-2), where i=min(t; n-(t-1)^2), j=min(t; t^2-n+1), t=floor(sqrt(n-1))+1.

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 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.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   9  16  25  36 ...
   7   8   6  11  18  27 ...
  13  14  12  15  24  35 ...
  21  22  20  23  19  28 ...
  31  32  30  33  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  9,  8, 13;
  17, 16,  6, 14, 21;
  26, 25, 11, 12, 22, 31;
  ...

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. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

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

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+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).

cp's OEIS Frontend

A064788 Inverse permutation to A060736.

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A061349 Sum of antidiagonals of A060736.

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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.

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A220508 T(n,k) = n^2 + k if k <= n, otherwise T(n,k) = k*(k + 2) - n; square array T(n,k) read by ascending antidiagonals (n >= 0, k >= 0).

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A213921 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places clockwise. Table T(n,k) read by antidiagonals.

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A213922 Natural numbers placed in table T(n,k) layer by layer. The order of placement: T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1). Table T(n,k) read by antidiagonals.

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A194195 First inverse function (numbers of rows) for pairing function A060734.

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