A003057 -id:A003057 - cp's OEIS frontend

A213197 T(n,k) = (2(n+k)^2 - 2(n+k) - 4k + 6 + (2k-2)(-1)^n + (2k-1)(-1)^k + (-2n+1)*(-1)^(n+k))/4; n, k > 0, read by antidiagonals.

1, 3, 4, 2, 6, 5, 8, 9, 11, 12, 7, 15, 10, 14, 13, 17, 18, 20, 21, 23, 24, 16, 28, 19, 27, 22, 26, 25, 30, 31, 33, 34, 36, 37, 39, 40, 29, 45, 32, 44, 35, 43, 38, 42, 41, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 46, 66, 49, 65, 52, 64, 55, 63, 58, 62, 61, 68

Offset: 1

Boris Putievskiy, Mar 01 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.
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(1,2*m+1), T(1,2*m), T(2, 2*m-1), T(3, 2*m-1),... T(2*m,1), T(2*m+1,1);
T(2*m,2), T(2*m-2,4), ...T(2,2*m);
...
Movement along two adjacent antidiagonals. The first row consists of phases: step to the west, step to the southwest, step to the south. The second row consists of phases: 2 steps to the north, 2 steps to the east. The length of each step is 1.

			The start of the sequence as a table:
   1,  3,  2,  8,  7, 17, 16, ...
   4,  6,  9, 15, 18, 28, 31, ...
   5, 11, 10, 20, 19, 33, 32, ...
  12, 14, 21, 27, 34, 44, 51, ...
  13, 23, 22, 36, 35, 53, 52, ...
  24, 26, 37, 43, 54, 64, 75, ...
  25, 39, 38, 56, 55, 77, 76, ...
  ...
The start of the sequence as a triangular array read by rows:
   1;
   3,  4;
   2,  6,  5;
   8,  9, 11, 12;
   7, 15, 10, 14, 13;
  17, 18, 20, 21, 23, 24;
  16, 28, 19, 27, 22, 26, 25;
  ...
The start of the sequence as an array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from row 2*r-2 of the triangular array above.
Last  2*r-1 numbers are from row 2*r-1 of the triangular array above.
   1;
   3,  4,  2,  6,  5;
   8,  9, 11, 12,  7, 15, 10, 14, 13;
  17, 18, 20, 21, 23, 24, 16, 28, 19, 27, 22, 26, 25;
  ...
Row r contains permutation of 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+5, 2*r*r-5*r+6, ..., 2*r*r-2*r+2, 2*r*r-2*r+1.

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. A002260, A004736, A003056, A003057.

Table T(n,k) contains: in rows A130883, A033816, A100037, A000384, A100038, A014106, A091823; in columns A001844, A142463, A090288, A139570, A046092, A051890, A059993, A097080, A181510, A137882, A152813.

T:=(n,k)->(2*(n+k)^2-2*(n+k)-4*k+6+(2*k-2)*(-1)^n+(2*k-1)*(-1)^k+(1-+2*n)*(-1)^(n+k))/4: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

T[n_, k_] := (2(n+k)^2 - 2(n+k) - 4k + 6 + (2k-2)(-1)^n + (2k-1)(-1)^k + (-2n+1)(-1)^(n+k))/4;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=(2*(t+2)**2-2*(t+2)-4*j+6 +(2*j-2)*(-1)**i+(2*j-1)*(-1)**j+(-2*i+1)*(-1)**t)/4

As a table:
T(n,k) = (2*(n+k)^2 - 2*(n+k) - 4*k + 6 + (2*k-2)*(-1)^n + (2*k-1)*(-1)^k + (-2*n+1)*(-1)^(n+k))/4.
As a linear sequence:
a(n) = (2*A003057(n)^2 - 2*A003057(n) - 4*A004736(n) + 6 + (2*A004736(n)-2)*(-1)^A002260(n) + (2*A004736(n)-1)*(-1)^A004736(n) + (-2*A002260(n)+1)*(-1)^A003056(n))/4;
a(n) = (2*(t+2)^2 - 2*(t+2) - 4*j + 6 + (2*j-2)*(-1)^i + (2*j-1)*(-1)^j + (-2*i+1)*(-1)^t)/4, where i = n - t*(t+1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2).

A213205 T(n,k) = ((k+n)^2-4k+3+(-1)^k-2(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 4, 2, 3, 6, 10, 9, 14, 13, 7, 8, 11, 12, 15, 19, 18, 23, 22, 27, 26, 16, 17, 20, 21, 24, 25, 28, 32, 31, 36, 35, 40, 39, 44, 43, 29, 30, 33, 34, 37, 38, 41, 42, 45, 49, 48, 53, 52, 57, 56, 61, 60, 65, 64, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 70

Offset: 1

Boris Putievskiy, Feb 15 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.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(1,3), T(2,2), T(2,1), T(1,2), T(3,1);
. . .
T(1,2*n+1), T(2,2*n), T(2,2*n-1), T(1,2*n), ...T(2*n-1,3), T(2*n,2), T(2*n,1), T(2*n-1,2), T(2*n+1,1);
. . .
Movement along two adjacent antidiagonals - step to the southwest, step to the west, step to the northeast, 2 steps to the south, step to the west and so on. The length of each step is 1.
Table contains:
row  1 accommodates elements A130883 in odd places,
row  2 is alternation of elements A100037 and A033816;
column 1 is alternation of elements A000384 and A091823,
column 2 is alternation of elements A014106 and A071355,
column 3 accommodates elements A130861 in even places;
main diagonal is alternation of elements A188135 and A033567,
diagonal 1, located above the main diagonal accommodates elements A033566 in even places,
diagonal 2, located above the main diagonal is alternation of elements A139271 and A024847,
diagonal 3, located above the main diagonal  accommodates of elements A033585.

			The start of the sequence as table:
1....5...2..10...7..19..16...
4....3...9...8..18..17..31...
6...14..11..23..20..36..33...
13..12..22..21..35..34..52...
15..27..24..40..37..57..54...
26..25..39..38..56..55..77...
28..44..41..61..58..82..79...
. . .
The start of the sequence as triangle array read by rows:
1;
5,4;
2,3,6;
10,9,14,13;
7,8,11,12,15;
19,18,23,22,27,26;
16,17,20,21,24,25,28;
. . .
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of  triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of  triangle array, located above.
1;
5,4,2,3,6;
10,9,14,13,7,8,11,12,15;
19,18,23,22,27,26,16,17,20,21,24,25,28;
. . .
Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+7, 2*r*r-5*r+6,...2*r*r-r-4, 2*r*r-r-3, 2*r*r-r.

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. A211377, A130883, A100037, A033816, A000384, A091823, A014106, A071355, A130861, A188135, A033567, A033566, A139271, A024847, A033585, A002260, A004736, A003056, A003057.

T:=(n,k)->((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

T[n_, k_] := ((n+k)^2 - 4k + 3 + (-1)^k - 2(-1)^n - (n+k)(-1)^(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

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)**2-4*j+3+(-1)**j-2*(-1)**i-(t+2)*(-1)**t)/2

As table
T(n,k) = ((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2.
As linear sequence
a(n) = (A003057(n)^2-4*A004736(n)+3+(-1)^A004736(n)-2*(-1)^A002260(n)-A003057(n)*(-1)^A003056(n))/2;
a(n) = ((t+2)^2-4*j+3+(-1)^j-2*(-1)^i-(t+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).

A349095 Where ones occur in A349082. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/x + 1/y, 0 < x < y.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 25, 26, 27, 31, 32, 33, 34, 38, 40, 41, 42, 43, 46, 47, 48, 49, 51, 59, 61, 63, 64, 65, 67, 68, 73, 80, 82, 83, 85, 86, 87, 94, 96, 97, 100, 101, 110, 113, 114, 115, 117, 121, 122, 123, 126, 129, 142, 143, 144, 145, 146, 147, 148

Offset: 1

Jud McCranie, Dec 24 2021

nonn

For index k, p/q = A002260(k)/A003057(k).

			6 is a term because A349082(6)=1, indicating that 3/4 = 1/x + 1/y has a unique solution, 1/2 + 1/4.

Jud McCranie, Table of n, a(n) for n = 1..500

Cf. A349082, A002260, A003057, A349090.

A381562 Minimum 2-tone chromatic number of maximal planar graphs with n vertices.

Original entry on oeis.org

6, 8, 9, 9, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 7

Offset: 3

Allan Bickle, Feb 27 2025

nonn

The 2-tone chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of vertices at distance 2 have two common colors.

For n in {19,22,23,27}, a(n) is either 7 or 8. All larger values are 7.

			For n=3, all 3 vertices get two distinct colors, so a(3) = 6.
For n=4, all 4 vertices get two distinct colors, so a(3) = 8.
For n=5 or 6, the extremal graph is a double wheel.

Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013) 171-190.
Allan Bickle, 2-Tone Coloring of Planar Graphs, Bull. Inst. Combin. Appl. 103 (2025) 114-129.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.

Cf. A003057, A351120 (pair coloring).

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles).

a(n) = 7 for n > 27.

A123572 The Kruskal-Macaulay function K_3(n).

Original entry on oeis.org

0, 3, 5, 6, 6, 8, 9, 9, 10, 10, 10, 12, 13, 13, 14, 14, 14, 15, 15, 15, 15, 17, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 23, 24, 24, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 28, 30, 31, 31, 32, 32, 32, 33, 33, 33, 33, 34, 34, 34, 34, 34

Offset: 0

N. J. A. Sloane, Nov 12 2006

Write n (uniquely) as n = C(n_t,t) + C(n_{t-1},t-1) + ... + C(n_v,v) where n_t > n_{t-1} > ... > n_v >= v >= 1. Then K_t(n) = C(n_t,t-1) + C(n_{t-1},t-2) + ... + C(n_v,v-1).

D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.

For K_i(n), i=1, 2, 3, 4, 5 see A000012, A003057, A123572, A123573, A123574.

lowpol := proc(n,t) local x::integer ; x := floor( (n*factorial(t))^(1/t)) ; while binomial(x,t) <= n do x := x+1 ; od ; RETURN(x-1) ; end: C := proc(n,t) local nresid,tresid,m,a ; nresid := n ; tresid := t ; a := [] ; while nresid > 0 do m := lowpol(nresid,tresid) ; a := [op(a),m] ; nresid := nresid - binomial(m,tresid) ; tresid := tresid-1 ; od ; RETURN(a) ; end: K := proc(n,t) local a ; a := C(n,t) ; add( binomial(op(i,a),t-i),i=1..nops(a)) ; end: A123572 := proc(n) K(n,3) ; end: for n from 0 to 80 do printf("%d, ",A123572(n)) ; od ; # R. J. Mathar, May 18 2007

lowpol[n_, t_] := Module[{x}, x = Floor[(n*t!)^(1/t)]; While[Binomial[x, t] <= n, x = x + 1]; x - 1];
c[n_, t_] := Module[{n0 = n, t0 = t, m, a = {}}, While[n0 > 0, m = lowpol[n0, t0]; a = Append[a, m]; n0 = n0 - Binomial[m, t0]; t0 = t0 - 1]; a];
K[n_, t_] := Module[{a}, a = c[n, t]; Sum[Binomial[a[[i]], t - i], {i, 1, Length[a]}]];
A123572[n_] := K[n, 3];
Table[A123572[n], {n, 0, 80}] (* Jean-François Alcover, Mar 22 2023, after R. J. Mathar *)

More terms from R. J. Mathar, May 18 2007

A123573 The Kruskal-Macaulay function K_4(n).

Original entry on oeis.org

0, 4, 7, 9, 10, 10, 13, 15, 16, 16, 18, 19, 19, 20, 20, 20, 23, 25, 26, 26, 28, 29, 29, 30, 30, 30, 32, 33, 33, 34, 34, 34, 35, 35, 35, 35, 38, 40, 41, 41, 43, 44, 44, 45, 45, 45, 47, 48, 48, 49, 49, 49, 50, 50, 50, 50, 52, 53, 53, 54, 54, 54, 55, 55, 55, 55, 56, 56, 56, 56, 56

Offset: 0

N. J. A. Sloane, Nov 12 2006

Write n (uniquely) as n = C(n_t,t) + C(n_{t-1},t-1) + ... + C(n_v,v) where n_t > n_{t-1} > ... > n_v >= v >= 1. Then K_t(n) = C(n_t,t-1) + C(n_{t-1},t-2) + ... + C(n_v,v-1).

D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.

For K_i(n), i=1, 2, 3, 4, 5 see A000012, A003057, A123572, A123573, A123574.

lowpol := proc(n,t) local x::integer ; x := floor( (n*factorial(t))^(1/t)) ; while binomial(x,t) <= n do x := x+1 ; od ; RETURN(x-1) ; end: C := proc(n,t) local nresid,tresid,m,a ; nresid := n ; tresid := t ; a := [] ; while nresid > 0 do m := lowpol(nresid,tresid) ; a := [op(a),m] ; nresid := nresid - binomial(m,tresid) ; tresid := tresid-1 ; od ; RETURN(a) ; end: K := proc(n,t) local a ; a := C(n,t) ; add( binomial(op(i,a),t-i),i=1..nops(a)) ; end: A123573 := proc(n) K(n,4) ; end: for n from 0 to 80 do printf("%d, ",A123573(n)) ; od ; # R. J. Mathar, May 18 2007

lowpol[n_, t_] := Module[{x}, x = Floor[(n*t!)^(1/t)]; While[Binomial[x, t] <= n, x = x + 1]; x - 1];
c[n_, t_] := Module[{n0 = n, t0 = t, m, a = {}}, While[n0 > 0, m = lowpol[n0, t0]; a = Append[a, m]; n0 = n0 - Binomial[m, t0]; t0 = t0 - 1]; a];
K[n_, t_] := Module[{a}, a = c[n, t]; Sum[Binomial[a[[i]], t - i], {i, 1, Length[a]}]];
A123573[n_] := K[n, 4];
Table[A123573[n], {n, 0, 70}] (* Jean-François Alcover, Mar 30 2023, after R. J. Mathar *)

More terms from R. J. Mathar, May 18 2007

A123574 The Kruskal-Macaulay function K_5(n).

Original entry on oeis.org

0, 5, 9, 12, 14, 15, 15, 19, 22, 24, 25, 25, 28, 30, 31, 31, 33, 34, 34, 35, 35, 35, 39, 42, 44, 45, 45, 48, 50, 51, 51, 53, 54, 54, 55, 55, 55, 58, 60, 61, 61, 63, 64, 64, 65, 65, 65, 67, 68, 68, 69, 69, 69, 70, 70, 70, 70, 74, 77, 79, 80, 80, 83, 85, 86, 86, 88, 89, 89, 90

Offset: 0

N. J. A. Sloane, Nov 12 2006

Write n (uniquely) as n = C(n_t,t) + C(n_{t-1},t-1) + ... + C(n_v,v) where n_t > n_{t-1} > ... > n_v >= v >= 1. Then K_t(n) = C(n_t,t-1) + C(n_{t-1},t-2) + ... + C(n_v,v-1).

D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.

For K_i(n), i=1, 2, 3, 4, 5 see A000012, A003057, A123572, A123573, A123574.

lowpol := proc(n,t) local x::integer ; x := floor( (n*factorial(t))^(1/t)) ; while binomial(x,t) <= n do x := x+1 ; od ; RETURN(x-1) ; end: C := proc(n,t) local nresid,tresid,m,a ; nresid := n ; tresid := t ; a := [] ; while nresid > 0 do m := lowpol(nresid,tresid) ; a := [op(a),m] ; nresid := nresid - binomial(m,tresid) ; tresid := tresid-1 ; od ; RETURN(a) ; end: K := proc(n,t) local a ; a := C(n,t) ; add( binomial(op(i,a),t-i),i=1..nops(a)) ; end: A123574 := proc(n) K(n,5) ; end: for n from 0 to 80 do printf("%d, ",A123574(n)) ; od ; # R. J. Mathar, May 18 2007

lowpol[n_, t_] := Module[{x}, x = Floor[(n*t!)^(1/t)]; While[Binomial[x, t] <= n, x = x + 1]; x - 1];
c[n_, t_] := Module[{n0 = n, t0 = t, m, a = {}}, While[n0 > 0, m = lowpol[n0, t0]; a = Append[a, m]; n0 = n0 - Binomial[m, t0]; t0 = t0 - 1]; a];
K[n_, t_] := Module[{a}, a = c[n, t]; Sum[Binomial[a[[i]], t - i], {i, 1, Length[a]}]];
A123574[n_] := K[n, 5];
Table[A123574[n], {n, 0, 69}] (* Jean-François Alcover, Mar 30 2023, after R. J. Mathar *)

More terms from R. J. Mathar, May 18 2007

A130829 2n+1 appears 2n times.

Original entry on oeis.org

3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17

Offset: 1

Paul Curtz, Jul 17 2007

Cf. A001670, A002024, A003057.

seq(2*n+1 $ 2*n, n=1..10); # Robert Israel, Jan 14 2015

from math import isqrt
def A130829(n): return 1|((m:=isqrt(n))+int((n-m*(m+1)<<2)>=1)<<1) # Chai Wah Wu, Oct 17 2022

a(n) = 2*floor(sqrt(n)+1/2)+1. - Mikael Aaltonen, Jan 14 2015
From Robert Israel, Jan 14 2015: (Start)
G.f.: (x/(1-x))*(1+2*Sum_{m>=0} x^(m*(m+1))) = (x/(1-x))*(1+x^(-1/4)*theta_2(0,x)) where theta_2 is the second Jacobi theta function.
a(2n) = a(2n-1) = 2*A002024(n)+1.
a(n) = A001670(n)+1.
(End)

A133196 n+2 repeated n times.

Original entry on oeis.org

3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15

Offset: 1

Paul Curtz, Oct 09 2007

nonn

Cf. A002024, A003056, A003057.

Table[PadRight[{},n,n+2],{n,20}]//Flatten (* Harvey P. Dale, Sep 04 2019 *)

from math import isqrt
def A133196(n): return 2+(isqrt(8*n)+1)//2 # Chai Wah Wu, Oct 17 2022

a(n) = A003056(n)+3, a(n) = A003057(n)+1; a(n) = t+3, where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, May 07 2013

A221215 T(n,k)= ((n+k)^2-2(n+k)+4-(n+3k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 6, 5, 2, 3, 4, 15, 14, 13, 12, 7, 8, 9, 10, 11, 28, 27, 26, 25, 24, 23, 16, 17, 18, 19, 20, 21, 22, 45, 44, 43, 42, 41, 40, 39, 38, 29, 30, 31, 32, 33, 34, 35, 36, 37, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 91

Offset: 1

Boris Putievskiy, Feb 22 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.
Enumeration table T(n,k). Let m be natural number. The order of the list:
T(1,1)=1;
T(1,3), T(2,2), T(3,1);
T(2,1), T(1,2);
. . .
T(1,2*m+1), T(2,2*m), T(3,2*m-1), ... T(2*m+1,1);
T(2*m,1), T(2*m-1,2), T(2*m-2,3),...T(1,2*m);
. . .
First row  contains elements antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read downwards.
second row contains elements antidiagonal {T(1,2*m), ... T(2*m,1)},  read upwards.
The same as A211394, except for reversed order in even diagonals. - M. F. Hasler, Feb 26 2013

			The start of the sequence as table:
1....6...2..15...7..28..16...
5....3..14...8..27..17..44...
4...13...9..26..18..43..31...
12..10..25..19..42..32..63...
11..24..20..41..33..62..50...
23..21..40..34..61..51..86...
22..39..35..60..52..85..73...
. . .
The start of the sequence as triangle array read by rows:
1;
6,5;
2,3,4;
15,14,13,12;
7,8,9,10,11;
28,27,26,25,24,23;
16,17,18,19,20,21,22;
. . .
Row number r contains r consecutive numbers.
If r is odd,  row is increasing.
If r is even, row is decreasing.

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. A211394, A002260, A004736, A003057.

T[n_, k_] := ((n+k)^2 - 2(n+k) + 4 - (n+3k-2)(-1)^(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 05 2019 *)

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*(t+2)+4-(i+3*j-2)*(-1)**t)/2

As table
T(n,k)= ((n+k)^2-2*(n+k)+4-(n+3*k-2)*(-1)^(n+k))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A003057(n)+4-(A002260(n)+3*A004736(n)-2)*(-1)^A003056(n))/2;  a(n) = ((t+2)^2-2*(t+2)+4-(i+3*j-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).

cp's OEIS Frontend

A213197 T(n,k) = (2*(n+k)^2 - 2*(n+k) - 4*k + 6 + (2*k-2)*(-1)^n + (2*k-1)*(-1)^k + (-2*n+1)*(-1)^(n+k))/4; n, k > 0, read by antidiagonals.

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A213205 T(n,k) = ((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.

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A349095 Where ones occur in A349082. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/x + 1/y, 0 < x < y.

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A381562 Minimum 2-tone chromatic number of maximal planar graphs with n vertices.

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A123572 The Kruskal-Macaulay function K_3(n).

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A123573 The Kruskal-Macaulay function K_4(n).

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A123574 The Kruskal-Macaulay function K_5(n).

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A130829 2n+1 appears 2n times.

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A213197 T(n,k) = (2(n+k)^2 - 2(n+k) - 4k + 6 + (2k-2)(-1)^n + (2k-1)(-1)^k + (-2n+1)*(-1)^(n+k))/4; n, k > 0, read by antidiagonals.

A213205 T(n,k) = ((k+n)^2-4k+3+(-1)^k-2(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.

A221215 T(n,k)= ((n+k)^2-2(n+k)+4-(n+3k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.