A130883 -id:A130883 - 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

A377137 Array read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n contains 3n/2 elements if n is even, and (n+1)/2 elements if n is odd; ; see Comments.

Original entry on oeis.org

1, 4, 2, 3, 6, 5, 12, 10, 8, 7, 9, 11, 15, 13, 14, 24, 22, 20, 18, 16, 17, 19, 21, 23, 28, 26, 25, 27, 40, 38, 36, 34, 32, 30, 29, 31, 33, 35, 37, 39, 45, 43, 41, 42, 44, 60, 58, 56, 54, 52, 50, 48, 46, 47, 49, 51, 53, 55, 57, 59, 66, 64, 62, 61, 63, 65, 84, 82, 80, 78, 76, 74, 72, 70, 68, 67, 69, 71, 73, 75, 77, 79, 81, 83, 91, 89, 87, 85, 86

Offset: 1

Boris Putievskiy, Oct 17 2024

Row n has length A064455(n). The sequence A064455 is non-monotonic.
The array consists of two triangular arrays alternating row by row.
For odd n, row n consists of permutations of the integers from A001844((n-1)/2) to A265225(n-1). For even n, row n consists of permutations of the integers from A130883(n/2) to A265225(n-1).
These permutations are generated by the algorithm described A130517.
The sequence is an intra-block permutation of the positive integers.

			Array begins:
     k =  1   2   3   4   5   6
  n=1:    1;
  n=2:    4,  2,  3;
  n=3:    6,  5;
  n=4:   12, 10,  8,  7,  9, 11;
The triangular arrays alternate by row: n=1 and n=3 comprise one, and n=2 and n=4 comprise the other.
Subtracting (n^2 - 1)/2 if n is odd from each term in row n produces a permutation of 1 .. (n+1)/2. Subtracting (n^2 - n)/2 if n is even from each term in row n produces a permutation of 1 .. 3n/2:
  1,
  3, 1, 2,
  2, 1,
  6, 4, 2, 1, 3, 5,
  ...

Boris Putievskiy, Table of n, a(n) for n = 1..9940
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A064455, A001844, A130517, A130883, A162630, A265225, A375602, A376180.

a[n_]:=Module[{L,R, P,Result},L=Ceiling[Max[x/.NSolve[x*(2*(x-1)-Cos[Pi*(x-1)]+5)-4*n==0,x,Reals]]]; R=n-If[EvenQ[L],(L^2-L)/2,(L^2-1)/2]; P[(L+1)*(2*L-(-1)^L+5)/4]=If[EvenQ[L],3L/2,(L+1)/2]; P[3]=2; P= Abs[2*R-If[EvenQ[L],3L/2,(L+1)/2]-If[2*R<=If[EvenQ[L],3L/2,(L+1)/2]+1,2,1]]; Res=P+If[EvenQ[L],(L^2-L)/2,(L^2-1)/2]; Result=Res; Result] Nmax= 12; Table[a[n],{n,1,Nmax}]

Linear sequence:
a(n) = P(n) + B(L(n)-1), where L(n) = ceiling(x(n)), x(n) is largest real root of the equation B(x) - n = 0. B(n) = (n+1)*(2*n-(-1)^n+5)/4 = A265225(n). P(n) = A162630(n)/2.
Array T(n,k) (see Example):
T(n, k) = P(n, k) + (n^2 - n)/2 if n is even, T(n, k) = P(n, k) + (n^2 - 1)/2 if n is odd, T(n, k) = P(n, k) +  A265225(n-1). P(n, k) = |2k - 3n / 2 - 2| if n is even and if 2k <= 3n / 2 + 1, P(n, k) = |2k - 3n / 2 - 1| if n is even and if 2k > 3n / 2 + 1. P(n, k) = |2k - (n + 1) / 2 - 2| if n is odd and if 2k <=  (n + 1) / 2 + 1, P(n, k) = |2k - (n + 1) / 2 - 1| if n is odd and if 2k > (n + 1) / 2 + 1. There are several special cases: P(n, 1) = 3n/2 if n is even, P(n, 1) = (n+1)/2 if n is odd. P(2, 2) = 1. P(n, n) = n/2 - 1 if n is even, P(n, n) = (n-3)/2 if n is odd.

A320530 T(n,k) = k^n + k^(n - 2)n(n - 1)(k(k - 1) + 1)/2 for 0 < k <= n and T(n,0) = A154272(n+1), square array read by antidiagonals upwards.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 7, 3, 1, 0, 7, 26, 16, 4, 1, 0, 11, 88, 90, 29, 5, 1, 0, 16, 272, 459, 220, 46, 6, 1, 0, 22, 784, 2133, 1504, 440, 67, 7, 1, 0, 29, 2144, 9234, 9344, 3775, 774, 92, 8, 1, 0, 37, 5632, 37908, 54016, 29375, 7992, 1246, 121, 9

Offset: 0

Franck Maminirina Ramaharo, Oct 14 2018

Construct a length n ternary word over the alphabet {a, b, c} as follows: letters from the set {a, b} are only used in pairs of at most one, and consist of either (a,b), (b,a) or (b,b). Next, replace each occurrence of a, b and c with a length k binary word such that 'a' has exactly two letters 1, 'b' contains no 0's and 'c' has exactly one letter 0 (empty words otherwise, respectively). Then T(n,k) gives the number of length n*k binary words resulting from this substitution. First column follows from the next definition.
In Kauffman's language, T(n,k) is the number of ways of splitting the crossings of the Pretzel knot shadow P(k, k, ..., k) having n tangles, of k half-twists respectively, such that the final diagram consists of two Jordan curves. This result can be achieved by assigning each tangle of the Pretzel knot a length k binary words in a way that letters 1 and 0 indicate the adequate choice for splitting the crossings.
Columns are linear recurrence sequences with signature (3*k, -3*k^2, k^3).

			Square array begins:
    1,  1,     1,     1,      1,       1,       1, ...
    0,  1,     2,     3,      4,       5,       6, ...
    1,  2,     7,    16,     29,      46,      67, ...
    0,  4,    26,    90,    220,     440,     774, ...
    0,  7,    88,   459,   1504,    3775,    7992, ...
    0, 11,   272,  2133,   9344,   29375,   74736, ...
    0, 16,   784,  9234,  54016,  212500,  649296, ...
    0, 22,  2144, 37908, 295936, 1456250, 5342112, ...
    ...
T(3,2) = 2^3 + 2^(3 - 2)*3*(3 - 1)*(2*(2 - 1) + 1)/2 = 26. The corresponding ternary words are abc, acb, cab, bac, bca, cba, bbc, bcb, cbb, ccc.  Next, let a = {00}, b = {11} and c = {01, 10}. The resulting binary words are
    abc: 001101, 001110;
    acb: 000111, 001011;
    cab: 010011, 100011;
    bac: 110001, 110010;
    bca: 110100, 111000;
    cba: 011100, 101100;
    bbc: 111101, 111110;
    bcb: 110111, 111011;
    cbb: 011111, 101111;
    ccc: 010101, 101010, 010110, 011001, 100101, 101001, 100110, 011010.

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Alexander Stoimenow, Everywhere Equivalent 2-Component Links, Symmetry Vol. 7 (2015), 365-375.
Wikipedia, Pretzel link

Column 1 is column 2 of A300453.
Column 2 is column 2 of A300184.
Cf. A300401, A303273, A320530.

T[n_, k_] = If[k > 0, k^n + k^(n - 2)*n*(n - 1)*(k*(k - 1) + 1)/2, If[k == 0 && (n == 0 || n == 1), 1, 0]];
Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 10}]//Flatten

t(n, k) := k^n + k^(n - 2)*binomial(n, 2)*(2*binomial(k, 2) + 1)$
u(n) := if n = 0 or n = 1 then 1 else 0$
T(n, k) := if k = 0 then u(n) else t(n,k)$
tabl(nn) := for n:0 thru 10 do print(makelist(T(n, k), k, 0, nn))$

T(n,k) = k^n + k^(n - 2)*binomial(n, 2)*(2*binomial(k, 2) + 1), k > 0.
T(n,k) = (3*k)*T(n-1,k) - (3*k^2)*T(n-2,k) + (k^3)*T(n-3,k), n > 3.
T(n,1) = A152947(n+1).
T(n,2) = A300451(n).
T(2,n) = A130883(n).
G.f. for columns: (1 - 2*k*x + (1 - k + 2*k^2)*x^2 )/(1 - k*x)^3.
E.g.f. for columns: ((1 - k + k^2)*x^2 + 2)*exp(k*x)/2.

A378626 Table T(n, k) read by upward antidiagonals. T(n,1) = A377137(n), T(n,2) = A377137(A377137(n)), T(n,3) = A377137(A377137(A377137(n))) and so on.

Original entry on oeis.org

1, 4, 1, 2, 3, 1, 3, 4, 2, 1, 6, 2, 3, 4, 1, 5, 5, 4, 2, 3, 1, 12, 6, 6, 3, 4, 2, 1, 10, 11, 5, 5, 2, 3, 4, 1, 8, 7, 9, 6, 6, 4, 2, 3, 1, 7, 10, 12, 8, 5, 5, 3, 4, 2, 1, 9, 12, 7, 11, 10, 6, 6, 2, 3, 4, 1, 11, 8, 11, 12, 9, 7, 5, 5, 4, 2, 3, 1, 15, 9, 10, 9, 11, 8, 12, 6, 6, 3, 4, 2, 1, 13, 14, 8, 7, 8, 9, 10, 11, 5, 5, 2, 3, 4, 1, 14, 15, 13, 10, 12, 10, 8, 7, 9, 6, 6, 4, 2

Offset: 1

Boris Putievskiy, Dec 02 2024

The sequence A377137 generates infinite cyclic group under composition. The identity element is A000027.
Each column is array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block.
Row n has length A064455(n). The sequence A064455 is non-monotonic.
The array consists of two triangular arrays alternating row by row.
For odd n, row n consists of permutations of the integers from A001844((n-1)/2) to A265225(n-1). For even n, row n consists of permutations of the integers from A130883(n/2) to A265225(n-1).
Each column is an intra-block permutation of the positive integers.

			Table begins:
  k =      1   2   3   4   5   6
--------------------------------------
  n =  1:  1,  1,  1,  1,  1,  1, ...
  n =  2:  4,  3,  2,  4,  3,  2, ...
  n =  3:  2,  4,  3,  2,  4,  3, ...
  n =  4:  3,  2,  4,  3,  2,  4, ...
  n =  5:  6,  5,  6,  5,  6,  5, ...
  n =  6:  5,  6,  5,  6,  5,  6, ...
  n =  7: 12, 11,  9,  8, 10,  7, ...
  n =  8: 10,  7, 12, 11,  9,  8, ...
  n =  9:  8, 10,  7, 12, 11,  9, ...
  n = 10:  7, 12, 11,  9,  8, 10, ...
  n = 11:  9,  8, 10,  7, 12, 11, ...
  n = 12: 11,  9,  8, 10,  7, 12, ...
  n = 13: 15, 14, 13, 15, 14, 13, ...
  n = 14: 13, 15, 14, 13, 15, 14, ...
  n = 15: 14, 13, 15, 14, 13, 15, ...
Column k = 1 contains the start of A377137. Ord(T(1,1),T(2,1), ... T(15,1)) = 6, ord(T(1,1),T(2,1), ... T(24,1)) = 18, ord(T(1,1),T(2,1), ... T(45,1)) = 90, ord(T(1,1),T(2,1), ... T(112,1)) = 1260, where ord is order of permutation.
The first 6 antidiagonals are:
  1;
  4, 1;
  2, 3, 1;
  3, 4, 2, 1;
  6, 2, 3, 4, 1;
  5, 5, 4, 2, 3, 1;

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A064455 (row lengths), A265225, A377137, A378127.

a[n_]:=Module[{L,R,P,Result},L=Ceiling[Max[x/.NSolve[x*(2*(x-1)-Cos[Pi*(x-1)]+5)-4*n==0,x,Reals]]];R=n-If[EvenQ[L],(L^2-L)/2,(L^2-1)/2]; P[(L+1)*(2*L-(-1)^L+5)/4]=If[EvenQ[L],3L/2,(L+1)/2];P[3]=2;P=Abs[2*R-If[EvenQ[L],3L/2,(L+1)/2]-If[2*R<=If[EvenQ[L],3L/2,(L+1)/2]+1,2,1]];Res=P+If[EvenQ[L],(L^2-L)/2,(L^2-1)/2];Result=Res;Result] (*A377137*)
composeSequence[a_,n_,k_]:=Nest[a,n,k]
Nmax=15;Kmax=6;T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Kmax}]

(T(1,k),T(2,k), ... T(A265225(n),k)) is permutation of the integers from 1 to A265225(n). (T(1,k),T(2,k), ... T(A265225(n),k)) = (T(1,1),T(2,1), ... T(A265225(n),1))^k.

A383464 a(n) = 8n^2 - 5n + 1.

Original entry on oeis.org

1, 4, 23, 58, 109, 176, 259, 358, 473, 604, 751, 914, 1093, 1288, 1499, 1726, 1969, 2228, 2503, 2794, 3101, 3424, 3763, 4118, 4489, 4876, 5279, 5698, 6133, 6584, 7051, 7534, 8033, 8548, 9079, 9626, 10189, 10768, 11363, 11974, 12601, 13244, 13903, 14578, 15269

Offset: 0

N. J. A. Sloane, Jun 26 2025

This is equal to A139272(n) + 1, but has its own entry because of an important geometrical interpretation.
Definition: A k-legged Wu is a pencil of k semi-infinite lines originating from a common point.
A 2-legged Wu is a long-legged V (see A130883), and a 3-legged Wu is a long-legged Wu as in A140064.
Theorem (David Cutler, Jonathan Pei, and Edward Xiong, Jun 24 2025): a(n) is the maximum number of regions in the plane that can be formed from n copies of a 4-legged Wu.
Proof: See "Cutting a pancake with an exotic knife".

David O. H. Cutler and Neil J. A. Sloane, Cutting a pancake with an exotic knife, Paper in preparation, Sep 05 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..3499
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A130083, A139272, A140064.

I:=[1, 4, 23]; [n le 3 select I[n] else 3*Self(n-1)-3* Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 27 2025

LinearRecurrence[{3,-3,1},{1,4,23},50] (* Vincenzo Librandi, Jun 27 2025 *)

G.f.: (1 + x + 14*x^2)/(1 - x)^3.

E.g.f.: exp(x)*(1 + 3*x + 8*x^2). - Stefano Spezia, Jun 30 2025

A185669 a(n) = 4n^2 + 3n + 2.

Original entry on oeis.org

2, 9, 24, 47, 78, 117, 164, 219, 282, 353, 432, 519, 614, 717, 828, 947, 1074, 1209, 1352, 1503, 1662, 1829, 2004, 2187, 2378, 2577, 2784, 2999, 3222, 3453, 3692, 3939, 4194, 4457, 4728, 5007, 5294, 5589, 5892, 6203, 6522, 6849, 7184, 7527, 7878, 8237, 8604, 8979, 9362, 9753, 10152, 10559, 10974, 11397, 11828

Offset: 0

Paul Curtz, Feb 09 2011

Natural numbers A000027 written clockwise as a square spiral:
.
  43--44--45--46--47--48--49
   |
  42  21--22--23--24--25--26
   |   |                   |
  41  20   7---8---9--10  27
   |   |   |           |   |
  40  19   6   1---2  11  28
   |   |   |       |   |   |
  39  18   5---4---3  12  29
   |   |               |   |
  38  17--16--15--14--13  30
   |                       |
  37--36--35--34--33--32--31
.
Walking in straight lines away from the center:
1,  2, 11, ... = A054552(n)     = 1 -3*n+4*n^2,
1,  8, 23, ... = A033951(n)     = 1 +3*n+4*n^2,
1,  3, 13, ... = A054554(n+1)   = 1 -2*n-4*n^2,
1,  7, 21, ... = A054559(n+1)   = 1 +2*n+4*n^2,
1,  4, 15, ... = A054556(n+1)   = 1   -n+4*n^2,
1,  6, 19, ... = A054567(n+1)   = 1   +n+4*n^2,
1,  5, 17, ... = A053755(n)     = 1     +4*n^2,
1,  9, 25, ... = A016754(n)     = 1 +4*n+4*n^2 = (1+2*n)^2,
2,  8, 22, ... = 2*A084849(n)   = 2 +2*n+4*n^2,
2, 12, 30, ... = A002939(n+1)   = 2 +6*n+4*n^2,
2,  9, 24, ... = a(n)           = 2 +3*n+4*n^2,
2, 10, 26, ... = A069894(n)     = 2 +4*n+4*n^2,
3, 11, 27, ... = A164897(n)     = 3 +4*n+4*n^2,
3, 12, 29, ... = A054552(n+1)+1 = 3 +5*n+4*n^2,
3, 14, 33, ... = A033991(n+1)   = 3 +7*n+4*n^2,
3, 15, 35, ... = A000466(n+1)   = 3 +8*n+4*n^2,
4, 14, 32, ... = 2*A130883(n+1) = 4 +6*n+4*n^2,
4, 16, 36, ... = A016742(n+1)   = 4 +8*n+4*n^2 = (2+2*n)^2,
5, 18, 39, ... = A007742(n+1)   = 5 +9*n+4*n^2,
5, 19, 41, ... = A125202(n+2)   = 5+10*n+4*n^2.

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

[2+3*n+4*n^2: n in [0..80]];  // Vincenzo Librandi, Feb 09 2011

Table[4n^2 + 3n + 2, {n,0,50}] (* G. C. Greubel, Jul 09 2017 *)
LinearRecurrence[{3,-3,1},{2,9,24},60] (* Harvey P. Dale, Aug 11 2021 *)

a(n)=4*n^2+3*n+2 \\ Charles R Greathouse IV, Apr 14 2014

a(n) = a(n-1) + 8*n - 1.
a(n) = 2*a(n-1) - a(n-2) + 8.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (2 +3*x +3*x^2)/(1-x)^3 . - R. J. Mathar, Feb 11 2011
a(n) = A033954(n) + 2. - Bruno Berselli, Apr 10 2011
E.g.f.: (4*x^2 + 7*x + 2)*exp(x). - G. C. Greubel, Jul 09 2017

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).

cp's OEIS Frontend

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

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A377137 Array read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n contains 3n/2 elements if n is even, and (n+1)/2 elements if n is odd; ; see Comments.

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A320530 T(n,k) = k^n + k^(n - 2)*n*(n - 1)*(k*(k - 1) + 1)/2 for 0 < k <= n and T(n,0) = A154272(n+1), square array read by antidiagonals upwards.

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A378626 Table T(n, k) read by upward antidiagonals. T(n,1) = A377137(n), T(n,2) = A377137(A377137(n)), T(n,3) = A377137(A377137(A377137(n))) and so on.

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A383464 a(n) = 8*n^2 - 5*n + 1.

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A185669 a(n) = 4*n^2 + 3*n + 2.

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A199855 Inverse permutation to A210521.

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

A320530 T(n,k) = k^n + k^(n - 2)n(n - 1)(k(k - 1) + 1)/2 for 0 < k <= n and T(n,0) = A154272(n+1), square array read by antidiagonals upwards.

A383464 a(n) = 8n^2 - 5n + 1.

A185669 a(n) = 4n^2 + 3n + 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.

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.