A130861 -id:A130861 - cp's OEIS frontend

A211377 T(n,k) = ((k + n)^2 - 4k + 3 + (-1)^k - (k + n - 2)(-1)^(k + n))/2; n, k > 0, read by antidiagonals.

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

Offset: 1

Boris Putievskiy, Feb 07 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(1,2), T(2,1), T(2,2), T(3,1);
...
T(1,n), T(1,n-1), T(2,n-2), T(2,n-1), T(3,n-2), T(3,n-3)...T(n,1);
...
Descent by snake along two adjacent antidiagonal - step to the west, step to the southwest, step to the east, step to the southwest and so on.  The length of each step is 1.
Table contains:
row 1 is alternation of elements A130883 and A033816,
row 2 accommodates elements A100037 in odd places;
column 1 is alternation of elements A000384 and A091823,
column 2 is alternation of elements A071355 and A014106,
column 3 accommodates elements A130861 in even places;
main diagonal accommodates elements A188135 in odd places,
diagonal 1, located above the main diagonal, is alternation of elements A033567 and A033566,
diagonal 2, located above the main diagonal, is alternation of elements A139271 and A033585.

			The start of the sequence as a table:
   1,  3,  2,   8,   7,  17,  16,  30,  29,  47,  46, ...
   4,  5,  9,  10,  18,  19,  31,  32,  48,  49,  69, ...
   6, 12, 11,  21,  20,  34,  33,  51,  50,  72,  71, ...
  13, 14, 22,  23,  35,  36,  52,  53,  73,  74,  98, ...
  15, 25, 24,  38,  37,  55,  54,  76,  75, 101, 100, ...
  26, 27, 39,  40,  56,  57,  77,  78, 102, 103, 131, ...
  28, 42, 41,  59,  58,  80,  79, 105, 104, 134, 133, ...
  43, 44, 60,  61,  81,  82, 106, 107, 135, 136, 168, ...
  45, 63, 62,  84,  83, 109, 108, 138, 137, 171, 170, ...
  64, 65, 85,  86, 110, 111, 139, 140, 172, 173, 209, ...
  66, 88, 87, 113, 112, 142, 141, 175, 174, 212, 211, ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   3,  4;
   2,  5,  6;
   8,  9, 12, 13;
   7, 10, 11, 14, 15;
  17, 18, 21, 22, 25, 26;
  16, 19, 20, 23, 24, 27, 28;
  30, 31, 34, 35, 38, 39, 42, 43;
  29, 32, 33, 36, 37, 40, 41, 44, 45;
  47, 48, 51, 52, 55, 56, 59, 60, 63, 64;
  46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66;
  ...
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 number 2*r-2 of the triangular array above.
Last  2*r-1 numbers are from row number 2*r-1 of the triangular array above.
   1;
   3,  4,  2,  5,  6;
   8,  9, 12, 13,  7, 10, 11, 14, 15;
  17, 18, 21, 22, 25, 26, 16, 19, 20, 23, 24, 27, 28;
  30, 31, 34, 35, 38, 39, 42, 43, 29, 32, 33, 36, 37, 40, 41, 44, 45;
  47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66;
  ...
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+5, 2*r*r-5*r+6,...2*r*r-r-4, 2*r*r-r-1, 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 W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A000384, A014106, A033566, A033567, A033585, A033816, A071355, A091823, A100037, A130861, A130883, A139271, A188135.

T[n_, k_] := ((k+n)^2 - 4k + 3 + (-1)^k - (k+n-2)(-1)^(k+n))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 29 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-t*(-1)**(t+2))/2

As a table:
T(n,k) = ((k + n)^2 - 4*k + 3 + (-1)^k - (k + n - 2)*(-1)^(k + n))/2.
As a linear sequence:
a(n) = ((t + 2)^2 - 4*j + 3 + (-1)^j - t*(-1)^t)/2, where j = (t*t + 3*t + 4)/2 - n and t = int((sqrt(8*n - 7) - 1)/ 2).

A139579 a(n) = 2n^2 + 15n.

Original entry on oeis.org

0, 17, 38, 63, 92, 125, 162, 203, 248, 297, 350, 407, 468, 533, 602, 675, 752, 833, 918, 1007, 1100, 1197, 1298, 1403, 1512, 1625, 1742, 1863, 1988, 2117, 2250, 2387, 2528, 2673, 2822, 2975, 3132, 3293, 3458, 3627, 3800, 3977, 4158, 4343, 4532, 4725, 4922, 5123, 5328, 5537

Offset: 0

Omar E. Pol, May 19 2008

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139578, A139580, A139581.

LinearRecurrence[{3,-3,1},{0,17,38},50] (* Stefano Spezia, Oct 21 2023 *)

a(n)=2*n^2+15*n \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 4*n + 13; a(0) = 0. - Vincenzo Librandi, Nov 24 2010
From Stefano Spezia, Oct 21 2023: (Start)
O.g.f.: x*(17 - 13*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(17 + 2*x). (End)
From Amiram Eldar, Nov 10 2023: (Start)
Sum_{n>=1} 1/a(n) = 182144/675675 - 2*log(2)/15.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/15 - Pi/30 + 67952/675675. (End)

A139576 a(n) = n(2n + 9).

Original entry on oeis.org

0, 11, 26, 45, 68, 95, 126, 161, 200, 243, 290, 341, 396, 455, 518, 585, 656, 731, 810, 893, 980, 1071, 1166, 1265, 1368, 1475, 1586, 1701, 1820, 1943, 2070, 2201, 2336, 2475, 2618, 2765, 2916, 3071, 3230, 3393, 3560, 3731, 3906

Offset: 0

Omar E. Pol, May 19 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139577, A139578, A139579, A139580, A139581.

Cf. A277979.

s=0;lst={s};Do[s+=n++ +11;AppendTo[lst, s], {n, 0, 7!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2008 *)
Table[Sum[(2*i + n - 1), {i, 4, n}], {n, 3, 45}] (* Zerinvary Lajos, Jul 11 2009 *)
Table[n(2n+9),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,26},50] (* Harvey P. Dale, Dec 18 2018 *)

a(n)=n*(2*n+9) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 + 9*n.
a(n) = a(n-1) + 4*n + 7 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(11 - 7*x)/(1-x)^3.
E.g.f.: exp(x)*x*(11 + 2*x).
a(n) = A277979(n)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139577 a(n) = n(2n + 11).

Original entry on oeis.org

0, 13, 30, 51, 76, 105, 138, 175, 216, 261, 310, 363, 420, 481, 546, 615, 688, 765, 846, 931, 1020, 1113, 1210, 1311, 1416, 1525, 1638, 1755, 1876, 2001, 2130, 2263, 2400, 2541, 2686, 2835, 2988, 3145, 3306, 3471, 3640, 3813, 3990

Offset: 0

Omar E. Pol, May 19 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139578, A139579, A139580, A139581.

s=0;lst={s};Do[s+=n++ +13;AppendTo[lst, s], {n, 0, 7!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2008 *)
Table[n(2n+11),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,13,30},50] (* Harvey P. Dale, Mar 17 2019 *)

a(n)=n*(2*n+11) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 + 11*n.
a(n) = a(n-1) + 4*n + 9 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(13 - 9*x)/(1-x)^3.
E.g.f.: exp(x)*x*(13 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139578 a(n) = n(2n + 13).

Original entry on oeis.org

0, 15, 34, 57, 84, 115, 150, 189, 232, 279, 330, 385, 444, 507, 574, 645, 720, 799, 882, 969, 1060, 1155, 1254, 1357, 1464, 1575, 1690, 1809, 1932, 2059, 2190, 2325, 2464, 2607, 2754, 2905, 3060, 3219, 3382, 3549, 3720, 3895, 4074, 4257, 4444, 4635, 4830, 5029

Offset: 0

Omar E. Pol, May 19 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139579, A139580, A139581.

s=0;lst={s};Do[s+=n++ +15;AppendTo[lst, s], {n, 0, 7!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2008 *)
Table[n(2n+13),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,15,34},50] (* Harvey P. Dale, Nov 22 2014 *)

a(n)=n*(2*n+13) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*n^2 + 13*n.
a(n) = a(n-1) + 4*n + 11 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(15 - 11*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(15 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139580 a(n) = n(2n + 17).

Original entry on oeis.org

0, 19, 42, 69, 100, 135, 174, 217, 264, 315, 370, 429, 492, 559, 630, 705, 784, 867, 954, 1045, 1140, 1239, 1342, 1449, 1560, 1675, 1794, 1917, 2044, 2175, 2310, 2449, 2592, 2739, 2890, 3045, 3204, 3367, 3534, 3705, 3880, 4059

Offset: 0

Omar E. Pol, May 19 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139578, A139579, A139581.

a[n_]:=Sum[4*i+15, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 04 2008 *)
Table[n(2n+17),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,19,42},50] (* Harvey P. Dale, Jul 07 2024 *)

a(n)=n*(2*n+17) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 + 17*n.
a(n) = a(n-1) + 4*n + 15; a(0) = 0. - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(19 - 15*x)/(1-x)^3.
E.g.f.: exp(x)*x*(19 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139581 a(n) = n(2n + 19).

Original entry on oeis.org

0, 21, 46, 75, 108, 145, 186, 231, 280, 333, 390, 451, 516, 585, 658, 735, 816, 901, 990, 1083, 1180, 1281, 1386, 1495, 1608, 1725, 1846, 1971, 2100, 2233, 2370, 2511, 2656, 2805, 2958, 3115, 3276, 3441, 3610, 3783, 3960, 4141

Offset: 0

Omar E. Pol, May 19 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139578, A139579, A139580.

Table[n(2n+19),{n,0,50}]  (* Harvey P. Dale, Jan 26 2011 *)

a(n)=n*(2*n+19) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 + 19*n.
a(n) = a(n-1) + 4*n + 17 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(21 - 17*x)/(1-x)^3.
E.g.f.: exp(x)*x*(21 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A185878 Accumulation array of A185877, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 11, 10, 3, 24, 28, 18, 4, 45, 60, 51, 28, 5, 76, 110, 108, 80, 40, 6, 119, 182, 195, 168, 115, 54, 7, 176, 280, 318, 300, 240, 156, 70, 8, 249, 408, 483, 484, 425, 324, 203, 88, 9, 340, 570, 696, 728, 680, 570, 420, 256, 108, 10, 451, 770, 963, 1040, 1015, 906, 735, 528, 315, 130, 11, 584, 1012, 1290, 1428, 1440, 1344, 1162, 920, 648, 380, 154, 12

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ...

See A144112 for the definition of accumulation array.

			Northwest corner:
  1,  4, 11,  24,  45, ...
  2, 10, 28,  60, 110, ...
  3, 18, 51, 108, 195, ...
  4, 28, 80, 168, 300, ...
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185877, A185879.
Row 1 to 3: A006527, A006331, A064043.
Column 1 to 5: A000027, A028552, A140677, 12*A000096, 5*A130861.

f[n_, k_] := k*n*(2*k^2 - 3*k + 3*k*n - 3*n + 7)/6; Table[f[n - k + 1, k], {n,10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(n,k) = k*n*(2*k^2 -3*k +3*k*n -3*n +7)/6, k>=1, n>=1.

A211394 T(n,k) = (k+n)(k+n-1)/2-(k+n-1)(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 6, 2, 3, 4, 12, 13, 14, 15, 7, 8, 9, 10, 11, 23, 24, 25, 26, 27, 28, 16, 17, 18, 19, 20, 21, 22, 38, 39, 40, 41, 42, 43, 44, 45, 29, 30, 31, 32, 33, 34, 35, 36, 37, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 80

Offset: 1

Boris Putievskiy, Feb 08 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(3,1);
T(1,2), T(2,1);
. . .
T(1,n), T(2,n-1), T(3,n-2), ... T(n,1);
T(1,n-1), T(2,n-3), T(3,n-4),...T(n-1,1);
. . .
First row  matches with the elements antidiagonal {T(1,n), ... T(n,1)},
second row matches with the elements antidiagonal {T(1,n-1), ... T(n-1,1)}.
Table contains:
row  1 is alternation of elements A130883 and A096376,
row  2  accommodates elements A033816 in even places,
row  3  accommodates elements A100037 in odd  places,
row  5  accommodates elements A100038 in odd  places;
column 1 is alternation of elements A084849 and  A000384,
column 2 is alternation of elements A014106 and  A014105,
column 3 is alternation of elements A014107 and  A091823,
column 4 is alternation of elements A071355 and |A168244|,
column 5 accommodates elements A033537 in even places,
column 7 is alternation of elements A100040 and  A130861,
column 9 accommodates elements A100041 in even places;
the main diagonal is A058331,
diagonal  1, located above the main diagonal is  A001844,
diagonal  2, located above the main diagonal is  A001105,
diagonal  3, located above the main diagonal is  A046092,
diagonal  4, located above the main diagonal is  A056220,
diagonal  5, located above the main diagonal is  A142463,
diagonal  6, located above the main diagonal is  A054000,
diagonal  7, located above the main diagonal is  A090288,
diagonal  9, located above the main diagonal is  A059993,
diagonal 10, located above the main diagonal is |A147973|,
diagonal 11, located above the main diagonal is  A139570;
diagonal  1, located under the main diagonal is  A051890,
diagonal  2, located under the main diagonal is  A005893,
diagonal  3, located under the main diagonal is  A097080,
diagonal  4, located under the main diagonal is  A093328,
diagonal  5, located under the main diagonal is  A137882.

			The start of the sequence as table:
  1....5...2..12...7..23..16...
  6....3..13...8..24..17..39...
  4...14...9..25..18..40..31...
  15..10..26..19..41..32..60...
  11..27..20..42..33..61..50...
  28..21..43..34..62..51..85...
  22..44..35..63..52..86..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  5,6;
  2,3,4;
  12,13,14,15;
  7,8,9,10,11;
  23,24,25,26,27,28;
  16,17,18,19,20,21,22;
  . . .
Row number r matches with r numbers segment {(r+1)*r/2-r*(-1)^(r+1)-r+2,... (r+1)*r/2-r*(-1)^(r+1)+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. A130883, A096376, A033816, A100037, A100038, A084849, A000384, A014106, A014105, A014107, A091823, A071355, A168244, A033537, A100040, A130861, A100041, A058331, A001844, A001105, A046092, A056220, A142463, A054000, A090288, A059993, A147973, A139570, A051890, A005893, A097080, A093328, A137882.

T[n_, k_] := (n+k)(n+k-1)/2 - (-1)^(n+k)(n+k-1) - 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)
j=(t*t+3*t+4)/2-n
result=(t+2)*(t+1)/2-(t+1)*(-1)**t-j+2

T(n,k) = (k+n)*(k+n-1)/2-(k+n-1)*(-1)^(k+n)-k+2.
As linear sequence
a(n) = A003057(n)*A002024(n)/2- A002024(n)*(-1)^A003056(n)-A004736(n)+2.
a(n) = (t+2)*(t+1)/2 - (t+1)*(-1)^t-j+2, where j=(t*t+3*t+4)/2-n and t=int((math.sqrt(8*n-7) - 1)/ 2).

A213171 T(n,k) = ((k+n)^2 - 4k + 3 - (-1)^n - (k+n)(-1)^(k+n))/2; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 4, 5, 2, 3, 6, 9, 10, 13, 14, 7, 8, 11, 12, 15, 18, 19, 22, 23, 26, 27, 16, 17, 20, 21, 24, 25, 28, 31, 32, 35, 36, 39, 40, 43, 44, 29, 30, 33, 34, 37, 38, 41, 42, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 69

Offset: 1

Boris Putievskiy, Feb 14 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(1,2), T(2,1), T(3,1);
. . .
T(1,n), T(2,n-1), T(1,n-1), T(2,n-2), T(3,n-2), T(4,n-3)...T(n,1);
...
Descent by snake along two adjacent antidiagonals - step to the southwest, step to the north, step to the southwest, step to the south and so on. The length of each step is 1. Phase four steps is rotated 90 degrees counterclockwise and the mirror of the phase A211377.
Table contains the following:
row 1 is alternation of elements A130883 and A100037,
row 2 accommodates elements A033816 in even places;
column 1 is alternation of elements A000384 and A014106,
column 2 is alternation of elements A091823 and A071355,
column 4 accommodates elements A130861 in odd places;
main diagonal is alternation of elements A188135 and A033567,
diagonal 1, located above the main diagonal, accommodates elements A033585 in even places,
diagonal 2, located above the main diagonal, accommodates elements A139271 in odd places,
diagonal 3, located above the main diagonal, is alternation of elements A033566 and A194431.

			The start of the sequence as a table:
   1   4   2   9   7   8  16 ...
   5   3  10   8  19  17  32 ...
   6  13  11  22  20  35  33 ...
  14  12  23  21  36  34  53 ...
  15  26  24  39  37  56  54 ...
  27  25  40  38  57  55  78 ...
  28  43  41  60  58  81  79 ...
  ...
The start of the sequence as a triangle array read by rows:
   1
   4  5
   2  3  6
   9 10 13 14
   7  8 11 12 15
  18 19 22 23 26 27
  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
   4  5  2  3  6
   9 10 13 14  7  8 11 12 15
  18 19 22 23 26 27 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+6, 2*r*r-5*r+7, ..., 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. A000384, A002260, A003056, A003057, A004736, A014106, A033566, A033567, A033585, A033816, A071355, A091823, A100037, A130861, A130883, A139271, A188135, A194431, A211377.

T:=(n,k)->((k+n)^2-4*k+3-(-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)^n - (-1)^(n+k)(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)**i-(t+2)*(-1)**t)/2

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

cp's OEIS Frontend

A211377 T(n,k) = ((k + n)^2 - 4*k + 3 + (-1)^k - (k + n - 2)*(-1)^(k + n))/2; n, k > 0, read by antidiagonals.

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A139579 a(n) = 2*n^2 + 15*n.

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A139576 a(n) = n*(2*n + 9).

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A139577 a(n) = n*(2*n + 11).

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A139578 a(n) = n*(2*n + 13).

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A139580 a(n) = n*(2*n + 17).

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A139581 a(n) = n*(2*n + 19).

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A185878 Accumulation array of A185877, by antidiagonals.

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A211377 T(n,k) = ((k + n)^2 - 4k + 3 + (-1)^k - (k + n - 2)(-1)^(k + n))/2; n, k > 0, read by antidiagonals.

A139579 a(n) = 2n^2 + 15n.

A139576 a(n) = n(2n + 9).

A139577 a(n) = n(2n + 11).

A139578 a(n) = n(2n + 13).

A139580 a(n) = n(2n + 17).

A139581 a(n) = n(2n + 19).

A211394 T(n,k) = (k+n)(k+n-1)/2-(k+n-1)(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.

A213171 T(n,k) = ((k+n)^2 - 4k + 3 - (-1)^n - (k+n)(-1)^(k+n))/2; n, k > 0, read by antidiagonals.