A033816 -id:A033816 - cp's OEIS frontend

A316224 a(n) = n(2n + 1)(4n + 1).

0, 15, 90, 273, 612, 1155, 1950, 3045, 4488, 6327, 8610, 11385, 14700, 18603, 23142, 28365, 34320, 41055, 48618, 57057, 66420, 76755, 88110, 100533, 114072, 128775, 144690, 161865, 180348, 200187, 221430, 244125, 268320, 294063, 321402, 350385, 381060, 413475, 447678, 483717

Offset: 0

Bruno Berselli, Jun 27 2018

Sums of the consecutive integers from A000384(n) to A000384(n+1)-1. This is the case s=6 of the formula n*(n*(s-2) + 1)*(n*(s-2) + 2)/2 related to s-gonal numbers.

The inverse binomial transform is 0, 15, 60, 48, 0, ... (0 continued).

			Row sums of the triangle:
|  0 |  ................................................................. 0
|  1 |  2  3  4  5  .................................................... 15
|  6 |  7  8  9 10 11 12 13 14  ........................................ 90
| 15 | 16 17 18 19 20 21 22 23 24 25 26 27  ........................... 273
| 28 | 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  ............... 612
| 45 | 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65  .. 1155
...
where:
. first column is A000384,
. second column is A130883 (without 1),
. third column is A033816,
. diagonal is A014106,
. 0, 2, 8, 18, 32, 50, ... are in A001105.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First bisection of A059270 and subsequence of A034828, A047866, A109900, A290168.

Sums of the consecutive integers from P(s,n) to P(s,n+1)-1, where P(s,k) is the k-th s-gonal number: A027480 (s=3), A055112 (s=4), A228888 (s=5).

GAP
```
List([0..40], n -> n*(2*n+1)*(4*n+1));
```

[n*(2*n+1)*(4*n+1) for n in 0:40] |> println

Magma
```
[n*(2*n+1)*(4*n+1): n in [0..40]];
```

seq(n*(2*n+1)*(4*n+1),n=0..40); # Muniru A Asiru, Jun 27 2018

Table[n (2 n + 1) (4 n + 1), {n, 0, 40}]

Maxima
```
makelist(n*(2*n+1)*(4*n+1), n, 0, 40);
```
PARI
```
vector(40, n, n--; n*(2*n+1)*(4*n+1))
```
Python
```
[n*(2*n+1)*(4*n+1) for n in range(40)]
```
Sage
```
[n*(2*n+1)*(4*n+1) for n in (0..40)]
```

O.g.f.: 3*x*(5 + 10*x + x^2)/(1 - x)^4.
E.g.f.: x*(15 + 30*x + 8*x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) =  3*A258582(n).
a(n) = -3*A100157(-n).
Sum_{n>0} 1/a(n) = 2*(3 - log(4)) - Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 2*sqrt(2)*log(1+sqrt(2)) + (sqrt(2)-1/2)*Pi - 6. - Amiram Eldar, Sep 17 2022

A159938 The number of homogeneous trisubstituted linear alkanes.

Original entry on oeis.org

2, 6, 16, 36, 70, 122, 196, 296, 426, 590, 792, 1036, 1326, 1666, 2060, 2512, 3026, 3606, 4256, 4980, 5782, 6666, 7636, 8696, 9850, 11102, 12456, 13916, 15486, 17170, 18972, 20896, 22946, 25126, 27440, 29892, 32486

Offset: 2

Parthasarathy Nambi, Apr 26 2009

See the paper by Valentin Vankov Iliev for details.

This sequence is related to A152947 by a(n) = (n-1)*A152947(n) + sum( A152947(i), i=1..n-1 ). - Bruno Berselli, Dec 19 2013

			The number of homogeneous trisubstituted linear alkane with ten carbon atoms is 426.

Valentin Vankov Iliev, A mathematical characterization of the groups of substitution isomerism of the linear alkanes, J. Math. Chem. 47 (2010), 52-61.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002522, A033816, A081489, A152947, A159940, A159941.

a(n) = (1/3)*(2*n^3 - 9*n^2 + 19*n - 12), where n is the number of carbons.
a(n) = 2*A081489(n-1) = (n-1)*(2*n^2-7*n+12)/3. - R. J. Mathar, Apr 28 2009
G.f.: 2*x^2*(1-x+2*x^2)/(1-x)^4. - Colin Barker, Aug 06 2012

More terms from Colin Barker, Aug 06 2012

A159940 The number of trisubstitution products with composition C_n H_(2n-1) X_2 Y.

Original entry on oeis.org

4, 16, 46, 106, 208, 364, 586, 886, 1276, 1768, 2374, 3106, 3976, 4996, 6178, 7534, 9076, 10816, 12766, 14938, 17344, 19996, 22906, 26086, 29548, 33304, 37366, 41746, 46456, 51508, 56914, 62686, 68836, 75376, 82318, 89674, 97456, 105676, 114346

Offset: 2

Parthasarathy Nambi, Apr 26 2009

See the paper by Valentin Vankov Iliev for details.

			The number of trisubstitution products with composition C_n H_(2n-1) X_2 Y for n = 10 is 1276.

Valentin Vankov Iliev, A mathematical characterization of the groups of substitution isomerism of the linear alkanes, J. Math. Chem. 47 (2010), 52-61.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002522, A033816, A159938, A159941.

a(n) = (2 n^3 - 9 n^2 + 19 n - 14) where n is the number of carbons.

G.f.: 2*x^2*(2+3*x^2+x^3)/(x-1)^4. - R. J. Mathar, Apr 28 2009

More terms from R. J. Mathar, Apr 28 2009

A159941 Number of trisubstituted linear alkanes of composition C_n H_(2n-1) XYZ.

Original entry on oeis.org

8, 32, 92, 212, 416, 728, 1172, 1772, 2552, 3536, 4748, 6212, 7952, 9992, 12356, 15068, 18152, 21632, 25532, 29876, 34688, 39992, 45812, 52172, 59096, 66608, 74732, 83492, 92912, 103016, 113828, 125372, 137672

Offset: 2

Parthasarathy Nambi, Apr 26 2009

See the paper by Valentin Vankov Iliev for details.

			The number of trisubstituted linear decane (n = 10) is 2552.

V. V. Iliev, A mathematical characterization of the groups of substitution isomerism of the linear alkanes, J. Math. Chem. 47 (2010), p 52-61.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002522, A033816, A159938, A159940.

a(n) = (4*n^3 - 18*n^2 + 38*n - 28) where n is the number of carbons.
a(n) = 2*A159940(n). - R. J. Mathar, Jul 08 2009
G.f.: 4*x^2*(2+3*x^2+x^3)/(1-x)^4. - Colin Barker, Aug 07 2012
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, May 28 2021

A199855 Inverse permutation to A210521.

Original entry on oeis.org

1, 4, 2, 5, 3, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42, 34, 43, 35, 44, 36, 45, 56, 46, 57, 47, 58, 48, 59, 49, 60, 50, 61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66, 79

Offset: 1

Boris Putievskiy, Feb 04 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(2,1), T(2,2), T(1,2), T(1,3), T(3,1),
  ...
  T(2,n-1), T(4,n-3), T(6,n-5), ..., T(n,1),
  T(2,n),   T(4,n-2), T(6,n-4), ..., T(n,2),
  T(1,n),   T(3,n-2), T(5,n-4), ..., T(n-1,2),
  T(1,n+1), T(3,n-1), T(5,n-3), ..., T(n+1,1),
  ...
The order of the list elements of adjacent antidiagonals. Let m be a positive integer.
Movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(2,2*m-1) to T(2*m,1)   length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(2,2*m)   to T(2*m,2)   length of step is 2,
movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(1,2*m)   to T(2*m-1,2) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(1,2*m+1) to T(2*m+1,1) length of step is 2.
Table contains:
  row  1 is alternation of elements A001844 and A084849,
  row  2 is alternation of elements A130883 and A058331,
  row  3 is alternation of elements A051890 and A096376,
  row  4 is alternation of elements A033816 and A005893,
  row  6 is alternation of elements A100037 and A093328;
  row  5  accommodates elements A097080 in odd places,
  row  7  accommodates elements A137882 in odd places,
  row 10  accommodates elements A100038 in odd places,
  row 14  accommodates elements A100039 in odd places;
  column 1 is A093005 and alternation of elements A000384 and A001105,
  column 2 is alternation of elements A046092 and A014105,
  column 3 is A105638 and alternation of elements A014106 and A056220,
  column 4 is alternation of elements A142463 and A014107,
  column 5 is alternation of elements A091823 and A054000,
  column 6 is alternation of elements A090288 and |A168244|,
  column 8 is alternation of elements A059993 and A033537;
  column  7 accommodates elements  A071355  in odd  places,
  column  9 accommodates elements |A147973| in even places,
  column 10 accommodates elements  A139570  in odd  places,
  column 13 accommodates elements  A130861  in odd  places.

			The start of the sequence as table:
   1,  4,  5,  11,  13,  22,  25,  37,  41,  56,  61, ...
   2,  3,  7,   9,  16,  19,  29,  33,  46,  51,  67, ...
   6, 12, 14,  23,  26,  38,  42,  57,  62,  80,  86, ...
   8, 10, 17,  20,  30,  34,  47,  52,  68,  74,  93, ...
  15, 24, 27,  39,  43,  58,  63,  81,  87, 108, 115, ...
  18, 21, 31,  35,  48,  53,  69,  75,  94, 101. 123, ...
  28, 40, 44,  59,  64,  82,  88, 109, 116, 140, 148, ...
  32, 36, 49,  54,  70,  76,  95, 102, 124, 132, 157, ...
  45, 60, 65,  83,  89, 110, 117, 141, 149, 176, 185, ...
  50, 55, 71,  77,  96, 103, 125, 133, 158, 167, 195, ...
  66, 84, 90, 111, 118, 142, 150, 177, 186, 216, 226, ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   4,  2;
   5,  3,  6;
  11,  7, 12,  8;
  13,  9, 14, 10, 15;
  22, 16, 23, 17, 24, 18;
  25, 19, 26, 20, 27, 21, 28;
  37, 29, 38, 30, 39, 31, 40, 32;
  41, 33, 42, 34, 43, 35, 44, 36, 45;
  56, 46, 57, 47, 58, 48, 59, 49, 60, 50;
  61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66;
  ...
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, 2, 5, 3, 6;
  11, 7,12, 8,13, 9,14,10,15;
  22,16,23,17,24,18,25,19,26,20,27,21,28;
  37,29,38,30,39,31,40,32,41,33,42,34,43,35,44,36,45;
  56,46,57,47,58,48,59,49,60,50,61,51,62,52,63,53,64,54,65,55,66;
  ...
Row number r contains permutation numbers 4*r-3 from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-3*r+2,2*r*r-5*r+4, 2*r*r-3*r+3, 2*r*r-5*r+5, 2*r*r-3*r+4, 2*r*r-5*r+6, ..., 2*r*r-3*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 Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A210521, A001844, A084849, A130883, A058331, A051890, A096376, A033816, A005893, A100037, A093328, A097080, A137882, A100038, A100039, A093005, A000384, A001105, A046092, A014105, A105638, A014106, A056220, A142463, A014107, A091823, A054000, A090288, A168244, A059993, A033537, A071355, A147973, A139570, A130861.

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*j**2+(4*i-5)*j+2*i**2-3*i+2+(2+(-1)**j)*((1-(t+1)*(-1)**i)))/4

T(n,k) = (2*k^2+(4*n-5)*k+2*n^2-3*n+2+(2+(-1)^k)*((1-(k+n-1)*(-1)^i)))/4.

a(n) = (2*j^2+(4*i-5)*j+2*i^2-3*i+2+(2+(-1)^j)*((1-(t+1)*(-1)^i)))/4, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((sqrt(8*n-7) - 1)/2).

A216248 T(n,k)=((n+k)^2-4k+3-(-1)^k-(n+k-2)(-1)^(n+k))/2-1, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4k+3-(-1)^k-(n+k-2)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

Original entry on oeis.org

1, 2, 5, 3, 4, 6, 7, 10, 11, 14, 8, 9, 12, 13, 15, 16, 19, 20, 23, 24, 27, 17, 18, 21, 22, 25, 26, 28, 29, 32, 33, 36, 37, 40, 41, 44, 30, 31, 34, 35, 38, 39, 42, 43, 45, 46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 66, 67

Offset: 1

Boris Putievskiy, Mar 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). Let m be natural number. The order of the list:
T(1,1)=1;
T(1,2), T(1,3), T(2,2), T(2,1), T(3,1);
. . .
T(1,2*m), T(1,2*m+1), T(2,2*m), T(2,2*m-1), T(3,2*m-2), ... T(2*m-1,2), T(2*m-1,3), T(2*m,2), T(2*m,1), T(2*m+1,1);
. . .
Movement along two adjacent antidiagonals - step to the east, step to the southwest, step to the west, step to the southwest and so on. The length of each step is 1.

			The start of the sequence as table:
1....2...3...7...8..16..17...
5....4..10...9..19..18..32...
6...11..12..20..21..33..34...
14..13..23..22..36..35..53...
15..24..25..37..38..54..55...
27..26..40..39..57..56..78...
28..41..42..58..59..79..80...
. . .
The start of the sequence as triangular array read by rows:
1;
2,5;
3,4,6;
7,10,11,14;
8,9,12,13,15;
16,19,20,23,24,27;
17,18,21,22,25,26,28;
. . .
The start of the sequence as array read by rows, the length of row number 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;
2,5,3,4,6;
7,10,11,14,8,9,12,13,15;
16,19,20,23,24,27,17,18,21,22,25,26,28;
. . .
Row number r contains permutation of the 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+4, 2*r*r-5*r+7, ... 2*r*r-r-2, 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]
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A213205, A213171, A213197, A210521; table T(n,k) contains: in rows A033816, A130883, A100037, A100038, A100039; in columns A000384, A071355, A014106, A091823, A130861.

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
if j==1 and (i%2)==1:
   result=result-1

As table
T(n,k)=((n+k)^2-4*k+3-(-1)^k-(n+k-2)*(-1)^(n+k))/2-1, if k=1 and (n mod 2)=1;
T(n,k)=((n+k)^2-4*k+3-(-1)^k-(n+k-2)*(-1)^(n+k))/2,   else.
As linear sequence
a(n)=((t+2)^2-4*j+3-(-1)^j-(t)*(-1)^t)/2 -1, if j=1 and (i mod 2)=1;
a(n)=((t+2)^2-4*j+3-(-1)^j-(t)*(-1)^t)/2,    else; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A216249 T(n,k) = ((n+k)^2-4k+3-2(-1)^n+(-1)^k-(n+k-4)(-1)^(n+k))/2-2, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4k+3-2(-1)^n+(-1)^k-(n+k-4)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n , k > 0.

Original entry on oeis.org

1, 3, 2, 4, 5, 6, 8, 7, 12, 11, 9, 10, 13, 14, 15, 17, 16, 21, 20, 25, 24, 18, 19, 22, 23, 26, 27, 28, 30, 29, 34, 33, 38, 37, 42, 41, 31, 32, 35, 36, 39, 40, 43, 44, 45, 47, 46, 51, 50, 55, 54, 59, 58, 63, 62, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 66, 68, 67, 72, 71, 76, 75, 80, 79, 84, 83, 88, 87

Offset: 1

Boris Putievskiy, Mar 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). Let m be natural number. The order of the list:
T(1,1)=1;
T(2,1), T(1,2), T(1,3), T(2,2), T(3,1);
. . .
T(2,2*m-1), T(1,2*m), T(1,2*m+1), T(2,2*m), T(2*m-3,4), ... T(2*m,1), T(2*m-1,2), T(2*m-1,3), T(2*m,2), T(2*m+1,1);
. . .
Movement along two adjacent antidiagonals - step to the northeast, step to the east, step to the southwest, 3 steps to the west, 2 steps to the south and so on.
The length of each step is 1.

			The start of the sequence as table:
   1   3  4    8   9  17  18...
   2   5  7   10  16  19  29...
   6  12  13  21  22  34  35...
  11  14  20  23  33  36  50...
  15  25  26  38  39  55  56...
  24  27  37  40  54  57  75...
  28  42  43  59  60  80  81...
  ...
The start of the sequence as triangular array read by rows:
   1;
   3,  2;
   4,  5,  6;
   8,  7, 12, 11;
   9, 10, 13, 14, 15;
  17, 16, 21, 20, 25, 24;
  18, 19, 22, 23, 26, 27, 28;
  ...
As an array read by rows, where the length of row number 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;
  3,   2,   4,   5,   6;
  8,   7,  12,  11,   9,  10,  13,  14,  15;
  17, 16,  21,  20,  25,  24,  18,  19,  22,  23,  26,  27,  28;
  ...
Row number r contains permutation of the 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+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 Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A213205, A213171, A213197, A210521; table T(n,k) contains: in rows A100037, A033816, A130883, A100039, A100038; in columns A000384, A071355, A091823, A014106.

T[n_, k_] := ((n+k)^2 - 4k + 3 - 2(-1)^n + (-1)^k - (n+k-4)(-1)^(n+k))/2 - 2Boole[k == 1 && OddQ[n]];
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 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
result=((t+2)**2-4*j+3+(-1)**j-2*(-1)**i-(t-2)*(-1)**t)/2
if j==1 and (i%2)==1:
   result=result-2

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

A216250 T(n,k) = ((n+k)^2-4k+3-2(-1)^n-(-1)^k-(n+k-4)(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1; T(n,k) = ((n+k)^2-4k+3-2(-1)^n-(-1)^k-(n+k-4)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 8, 11, 12, 10, 9, 14, 13, 15, 16, 17, 20, 21, 24, 25, 19, 18, 23, 22, 27, 26, 28, 29, 30, 33, 34, 37, 38, 41, 42, 32, 31, 36, 35, 40, 39, 44, 43, 45, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 49, 48, 53, 52, 57, 56, 61, 60, 65, 64, 66, 67

Offset: 1

Boris Putievskiy, Mar 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). Let m be natural number. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2), T(1,3), T(3,1);
  . . .
  T(1,2*m), T(2,2*m-1), T(2,2*m), T(1,2*m+1), T(3,2*m-2), ... T(2*m-1,2), T(2*m,1), T(2*m,2), T(2*m-1,3), T(2*m+1,1);
  . . .
Movement along two adjacent antidiagonals - step to the southwest, step east, step to the northeast, 3 steps to the west, 2 steps to the south and so on. The length of each step is 1.

			The start of the sequence as table:
  1....2...5...7..10..16..19...
  3....4...8...9..17..18..30...
  6...11..14..20..23..33..36...
  12..13..21..22..34..35..51...
  15..24..27..37..40..54..57...
  25..26..38..39..55..56..76...
  28..41..44..58..61..79..82...
  . . .
The start of the sequence as triangular array read by rows:
  1;
  2,3;
  5,4,6;
  7,8,11,12;
  10,9,14,13,15;
  16,17,20,21,24,25;
  19,18,23,22,27,26,28;
  . . .
The start of the sequence as array read by rows, with length of row r: 4*r-3:
First 2*r-2 numbers are from the row number 2*r-2 of above triangle array.
Last  2*r-1 numbers are from the row number 2*r-1 of above triangle array.
  1;
  2,3,5,4,6;
  7,8,11,12,10,9,14,13,15;
  16,17,20,21,24,25,19,18,23,22,27,26,28;
  . . .
Row number r contains permutation of the 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r: 2*r*r-5*r+4, 2*r*r-5*r+5, ...2*r*r-r-2, 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. A213205, A213171, A213197, A210521; table T(n,k) contains: in rows A130883, A033816, A100037, A100038, A100039; in columns A000384, A014106, A071355, A091823, A130861.

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
if j==1 and (i%2)==1:
   result=result-3

As table
T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1;
T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2,   else.
As linear sequence
a(n) = ((t+2)^2-4*j+3-2*(-1)^i-(-1)^j-(t-2)*(-1)^t)/2-3, if j=1 and (i mod 2)=1;
a(n) = ((t+2)^2-4*j+3-2*(-1)^i-(-1)^j-(t-2)*(-1)^t)/2,   else; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A221216 T(n,k) = ((n+k)^2-2(n+k)+4-(3n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

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

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(3,1), T(2,2), T(1,3);
  T(1,2), T(2,1);
  . . .
  T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(2,2*m), T(1,2*m+1);
  T(1,2*m), T(2,2*m-1), T(3,2*m-2),...T(2*m-1,2),T(2*m,1);
  . . .
First row  contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)},  read downwards.

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

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, A221215, A002260, A004736, A003057;
table T(n,k) contains: in rows A084849, A096376, A014105, A014107, A168244, A033537, A100040, A100041;
in columns A130883, A000384, A014106, A033816, A100037, A091823, A071355, A100038, A100039,A130861;
main diagonal and parallel diagonals are  A058331, A001844, A005893,A046092, A093328, A142463, A090288, A059993, A051890, A001105, A097080, A056220, A137882, A054000.

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

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

A221217 T(n,k) = ((n+k)^2-2n+3-(n+k-1)(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 6, 5, 4, 3, 2, 15, 14, 13, 12, 11, 10, 9, 8, 7, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 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(3,1), T(2,2), T(1,3);
  T(2,1), T(1,2);
  . . .
  T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(1,2*m+1);
  T(2*m,1), T(2*m-1,2), T(2*m-2,3),...T(1,2*m);
  . . .
First row  contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)},  read upwards.

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

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, A221215, A002260, A004736, A003057, A002024;
table T(n,k) contains: in rows A084849, A000384, A014106, A014105, A014107, A091823, A071355, A091823, A071355,  A100040, A130861, A100041;
in columns A130883, A096376, A033816, A100037, A100038, A100039;
main diagonal and parallel diagonals are A058331, A051890, A005893, A097080, A093328, A137882, A001844, A001105, A056220, A142463, A054000, A090288, A059993.

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

As table
T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A002260(n)+3-A002024(n)*(1+2*(-1)^A003056(n)))/2;
a(n) = ((t+2)^2-2*i+3-(t+1)*(1+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

A316224 a(n) = n*(2*n + 1)*(4*n + 1).

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A159938 The number of homogeneous trisubstituted linear alkanes.

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A159940 The number of trisubstitution products with composition C_n H_(2n-1) X_2 Y.

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A159941 Number of trisubstituted linear alkanes of composition C_n H_(2n-1) XYZ.

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A199855 Inverse permutation to A210521.

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A216248 T(n,k)=((n+k)^2-4*k+3-(-1)^k-(n+k-2)*(-1)^(n+k))/2-1, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4*k+3-(-1)^k-(n+k-2)*(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

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A216249 T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2-2, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n , k > 0.

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A316224 a(n) = n(2n + 1)(4n + 1).

A216248 T(n,k)=((n+k)^2-4k+3-(-1)^k-(n+k-2)(-1)^(n+k))/2-1, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4k+3-(-1)^k-(n+k-2)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

A216249 T(n,k) = ((n+k)^2-4k+3-2(-1)^n+(-1)^k-(n+k-4)(-1)^(n+k))/2-2, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4k+3-2(-1)^n+(-1)^k-(n+k-4)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n , k > 0.

A216250 T(n,k) = ((n+k)^2-4k+3-2(-1)^n-(-1)^k-(n+k-4)(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1; T(n,k) = ((n+k)^2-4k+3-2(-1)^n-(-1)^k-(n+k-4)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

A221216 T(n,k) = ((n+k)^2-2(n+k)+4-(3n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

A221217 T(n,k) = ((n+k)^2-2n+3-(n+k-1)(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.