A326407 -id:A326407 - cp's OEIS frontend

A326411 Triangle T(n,k) read by rows: T(n,k) = the number of ways of seating n people around a table for the second time so that k pairs are maintained. Reflected and rotated sequences are counted as one.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 5, 5, 0, 1, 3, 12, 15, 20, 9, 0, 1, 23, 70, 112, 91, 49, 14, 0, 1, 177, 544, 740, 640, 302, 96, 20, 0, 1, 1553, 4500, 6003, 4725, 2439, 747, 165, 27, 0, 1, 14963, 41740, 53585, 41420, 20810, 7076, 1550, 260, 35, 0, 1

Offset: 0

Witold Tatkiewicz, Aug 07 2019

Poulet (1919) arrives at this triangle of numbers by considering n-sided polygons whose vertices lie on a circle. Call a side of such a polygon simple if its endpoints are adjacent on the circle. Then T(n,k) is the number of such polygons with k simple sides. There is also a connection with A002464 (see that entry). - N. J. A. Sloane, Mar 08 2022
Definition requires "pairs" and for n=0 it is assumed that there is 1 way of seating 0 people around a table for the second time so that 0 pairs are maintained and 1 person forms only one pair with him/herself. Therefore T(0,0)=1, T(1,0)=0 and T(1,1)=1.
The weighted average of each row using k as weights converges to 2 for large n and appears to be given by (Sum_{k} k*T(n,k))/n! = 2/(n-1) + 2.

			Assuming the initial order was {1,2,3,4,5} (therefore 1 and 5 form a pair as the first and last persons are neighbors in the case of a round table) there are 5 sets of ways of seating them again so that 3 pairs are conserved: {1,2,3,5,4}, {2,3,4,1,5}, {3,4,5,2,1}, {4,5,1,3,2}, {5,1,2,4,3}. Since within each set we do not allow for circular symmetry (e.g., {1,2,3,5,4} and its rotation to form {2,3,5,4,1} are counted as one) nor reflection ({1,2,3,5,4} and {4,5,3,2,1} are also counted as one), the total number of ways is 5 and therefore T(5,3)=5.
Unfolded table with n individuals (rows) forming k pairs (columns):
    0    1    2    3    4    5    6    7
0   1
1   0    1
2   0    0    1
3   0    0    0    1
4   0    0    2    0    1
5   1    0    5    5    0    1
6   3   12   15   20    9    0   1
7  23   70  112   91   49   14   0   1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50; rows 0..17 from Witold Tatkiewicz)
P. Poulet, Query 4750, L'Intermédiaire des Mathématiciens, 26 (1919), 117-121. (Page 117)
P. Poulet, Query 4750, L'Intermédiaire des Mathématiciens, 26 (1919), 117-121. (Pages 118, 119)
P. Poulet, Query 4750, L'Intermédiaire des Mathématiciens, 26 (1919), 117-121. (Pages 120, 121)
Witold Tatkiewicz, link for java program.

Cf. A002816 (column k=0).
Row sums: A001710(n-1) = Sum_k T(n,k).
Cf. also A326390 (accounting for rotation and reflection symmetry), A326397 (disregards reflection symmetry but allows rotation), A326407 (disregards rotation symmetry but allows reflection).
See in addition A002464.

Java
```
See Links section
```

A326411 := proc(n,k)
    option remember;
    if k > n or k < 0 then
        0;
    elif k = n then
        1;
    elif k =0 then
        if n < 5 then
            0 ;
        elif n = 5 then
            1 ;
        elif n = 6 then
            3 ;
        elif n = 7 then
            23 ;
        else
            # Poulet eq (6) page 120, shifted n->n-2
            -(n^3-8*n^2+18*n-21)*procname(n-1,0)
            -4*(n^2-5*n)*procname(n-2,0)
            +2*(n^3-11*n^2+33*n-18)*procname(n-3,0)
            -(n^2-7*n+9)*procname(n-4,0)
            -(n^3-10*n^2+28*n-15)*procname(n-5,0) ;
            -%/(n^2-7*n+9) ;
        end if;
    elif n <= 3 then
        0;
    else
        # Poulet eq (3) page 119
        2*(n-k)*procname(n-1,k-1)/(n-1)+2*k*procname(n-1,k)/(n-1)
            +(k-2)*procname(n-2,k-2)/(n-2) - 2*(k-1)*procname(n-2,k-1)/(n-2) + k*procname(n-2,k)/(n-2) ;
        %*n/k ;
    end if;
end proc:
for n from 0 to 12 do
    for k from 0 to n do
        printf("%a ",A326411(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Mar 17 2022

T[n_, k_] := T[n, k] = Which[k > n || k < 0, 0, k == n, 1, k == 0, Which[n<5, 0, n == 5, 1, n == 6, 3, n == 7, 23, True,
    pc = -(n^3 - 8*n^2 + 18*n - 21)*T[n-1, 0]
      - 4*(n^2 - 5*n)*T[n - 2, 0]
      + 2*(n^3 - 11*n^2 + 33*n - 18)*T[n-3, 0]
      - (n^2 - 7*n + 9)*T[n-4, 0]
      - (n^3 - 10*n^2 + 28*n - 15)*T[n-5, 0];
    -pc/(n^2 - 7*n + 9)], n <= 3, 0, True,
   pc = 2*(n-k)*T[n-1, k-1]/(n-1) + 2*k*T[n-1, k]/(n-1) +
     (k - 2)*T[n-2, k-2]/(n-2) -
     2*(k-1)*T[n-2, k-1]/(n-2) + k*T[n-2, k]/(n-2);
    pc*n/k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 17 2023, after R. J. Mathar *)

Q(n,k)={k*subst(serlaplace(polcoef((1 - 2*x -x^2)/((1 + x)*(1 + (1 - y)*x + y*x^2)) + O(x^n), n-1)), y, k)}
row(n)={Vec(if(n<3, 1, (Q(n,y/(y-1))/2 + (-1)^n)*(y-1)^n), -n-1)} \\ Andrew Howroyd, Mar 01 2024

It appears that Poulet gives recurrences that generate the whole triangle. - N. J. A. Sloane, Mar 09 2022
T(n,n) = 1;
T(n,n-1) = 0 for n >= 1;
T(n,n-2) = n*(n-3)/2 for n >= 4 [Poulet];
T(n,n-3) = n*(n-4)*(2*n-7)/3 for n >= 4 [Poulet, corrected by N. J. A. Sloane, Mar 09 2022]
T(n,n-4) = (25/24)*n^4 + (23/12)*n^3 - (169/24)*n^2 + (85/12)*n - 3 for n > 5 (conjectured); [see Poulet]
T(n,n-5) = (26/15)*n^5 + (25/6)*n^4 - (83/6)*n^3 + (221/6)*n^2 - (299/10)*n + 13 for n > 5 (conjectured); [see Poulet]
T(n,n-6) = (707/240)*n^6 + (2037/240)*n^5 - (413/16)*n^4 + (2233/16)*n^3 - (2777/15)*n^2 + (3739/20)*n - 57 for n > 6 (conjectured). [See Poulet]

A326405 Minesweeper sequence of positive integers arranged on a 2D grid along ascending antidiagonals.

Original entry on oeis.org

3, -1, -1, 3, -1, 2, -1, 4, 4, 0, -1, 4, -1, 2, 0, 3, -1, 3, -1, 1, 0, 2, -1, 3, 3, 1, 1, 0, -1, 3, -1, 2, 2, 1, 2, 0, -1, 3, 3, 2, -1, 2, -1, 2, 0, 2, -1, 2, 3, 2, 3, 2, -1, 1, 0, 1, 2, 2, -1, 3, -1, 2, 2, 1, 1, 0, -1, 1, 1, 3, -1, 3, -1, 1, 2, 1, 2, 0

Offset: 1

Witold Tatkiewicz, Sep 26 2019

Map the positive integers on a 2D grid starting with 1 in top left corner and continue along increasing antidiagonals.
Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around it.
If n is the original number, a(n) is the number that replaces it.
This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.
a(n) < 5 (conjectured).
Set of n such that a(n) = 4 is unbounded (conjectured).

			Consider positive integers distributed on the plane along antidiagonals:
   1  2  4  7 11 16 ...
   3  5  8 12 17 ...
   6  9 13 18 ...
  10 14 19 ...
  15 20 ...
  21 ...
  ...
1 is not prime and in its adjacent grid cells there are 3 primes: 2, 3 and 5. Therefore a(1) = 3.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 4 primes: 2, 5, 7 and 13. Therefore a(8) = 4.
From _Michael De Vlieger_, Oct 01 2019: (Start)
Replacing n with a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:
  3  *  3  *  *  3  2  *  *  2  1  * ...
  *  *  4  4  *  *  3  3  *  2  1  2
  2  4  *  3  3  *  3  2  2  1  1  1
  .  2  *  3  2  2  3  *  3  1  1  *
  .  1  1  2  *  2  3  *  *  2  1  1
  .  1  1  2  3  *  3  3  *  3  1  .
  .  2  *  2  2  *  3  2  3  *  2  1
  .  2  *  2  1  1  2  *  2  1  3  *
  .  1  1  2  1  1  1  2  3  2  3  *
  .  1  1  2  *  2  1  1  *  *  2  2
  .  2  *  3  2  *  1  1  2  2  1  1
  .  2  *  3  2  2  1  1  1  1  .  1
   ...  (End)

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (150 antidiagonals).
Michael De Vlieger, Minesweeper-style graph read along original mapping, replacing -1 with a "mine", and 0 with blank space.
Michael De Vlieger, Square plot of a million terms read along original mapping, with black indicating a prime and levels of gray commensurate to a(n).
Witold Tatkiewicz, link for Java program
Wikipedia, Minesweeper game

Different arrangements of integers:
Cf. A326406 - triangle maze,
Cf. A326407 - square mapping,
Cf. A326408 - square maze,
Cf. A326409 - Hamiltonian path,
Cf. A326410 - Ulam's spiral.

Java
```
See Links section.
```

Block[{n = 12, s}, s = ArrayPad[Array[1 + PolygonalNumber[#1 + #2 - 1] - #2 &, {# + 1, # + 1}], 1] &@ n; Table[If[PrimeQ@ m, -1, Count[#, ?PrimeQ] &@ Union@ Map[s[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[s, m] + {##} - 2 &, {3, 3}]]], {m, PolygonalNumber@ n}]] (* _Michael De Vlieger, Sep 30 2019 *)

A326406 Minesweeper sequence of positive integers arranged on a 2D grid along a triangular maze.

Original entry on oeis.org

3, -1, -1, 2, -1, 3, -1, 4, 4, 1, -1, 3, -1, 3, 2, 1, -1, 3, -1, 3, 2, 1, -1, 2, 3, 2, 3, 1, -1, 3, -1, 2, 2, 1, 2, 1, -1, 2, 3, 1, -1, 3, -1, 3, 2, 1, -1, 2, 3, 2, 3, 2, -1, 2, 1, 0, 1, 2, -1, 3, -1, 2, 2, 1, 2, 1, -1, 2, 2, 1, -1, 3, -1, 3, 4, 0, 1, 1

Offset: 1

Witold Tatkiewicz, Oct 02 2019

Write positive integers on a 2D grid starting with 1 in the top left corner and continue along the triangular maze as in A056023.
Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around it.
n is replaced by a(n).
This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.
a(n) < 5 (conjectured).
Set of n such that a(n) = 4 is unbounded (conjectured).

			Consider positive integers placed on the plane along a triangular maze:
   1  2  6  7 15 16 ...
   3  5  8 14 17 ...
   4  9 13 18 ...
  10 12 19 ...
  11 20 ...
  21 ...
  ...
1 is not prime and in adjacent grid cells there are 3 primes: 2, 3 and 5. Therefore a(1) = 3.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 4 primes: 2, 5, 7 and 13. Therefore a(8) = 4.
Replacing n by a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:
  3  *  3  *  2  1  1  *  2  1  1  * ...
  *  *  4  3  *  3  3  3  *  2  2  2
  2  4  *  3  2  *  *  2  1  2  *  1
  1  3  *  3  2  3  3  2  1  1  1  2
  *  3  2  2  *  2  2  *  2  1  .  1
  2  *  1  1  3  *  3  2  *  2  1  1
  1  2  3  2  3  *  3  2  3  *  1  .
  1  2  *  *  3  2  2  *  2  1  2  2
  *  2  2  4  *  2  1  2  3  2  2  *
  1  1  .  2  *  3  1  1  *  *  2  3
  .  1  2  3  3  *  2  2  3  2  1  1
  1  2  *  *  2  1  2  *  1  .  .  1
...
In order to produce sequence graph is read along original mapping.

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (150 antidiagonals).
Michael De Vlieger, Minesweeper-style graph read along original mapping, replacing -1 with a "mine", and 0 with blank space.
Michael De Vlieger, Square plot of a million terms read along original mapping, with black indicating a prime and levels of gray commensurate to a(n).
Witold Tatkiewicz, Java program
Wikipedia, Minesweeper game

Cf. A056023 - plane mapping
Different arrangements of integers:
Cf. A326405 - antidiagonals,
Cf. A326407 - square mapping,
Cf. A326408 - square maze,
Cf. A326409 - Hamiltonian path,
Cf. A326410 - Ulam's spiral.

Java
```
// See Links section.
```

Block[{n = 12, s}, s = ArrayPad[Array[If[OddQ[#1 + #2], 1 + PolygonalNumber[#1 + #2 - 1] - #2, PolygonalNumber[#1 + #2 - 2] + #2] &, {# + 1, # + 1}], 1] &@ n; Table[If[PrimeQ@ m, -1, Count[#, ?PrimeQ] &@ Union@ Map[s[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[s, m] + {##} - 2 &, {3, 3}]]], {m, PolygonalNumber@ n}]] (* _Michael De Vlieger, Oct 02 2019 *)

A326408 Minesweeper sequence of positive integers arranged on a 2D grid along a square maze.

Original entry on oeis.org

2, -1, -1, 3, -1, 3, -1, 4, 3, 1, -1, 4, -1, 3, 4, 2, -1, 2, -1, 2, 3, 3, -1, 2, 1, 0, 2, 3, -1, 2, -1, 2, 2, 1, 3, 2, -1, 1, 1, 2, -1, 4, -1, 2, 3, 3, -1, 1, 0, 0, 2, 3, -1, 1, 1, 1, 3, 3, -1, 3, -1, 2, 2, 1, 0, 1, -1, 3, 3, 2, -1, 2, -1, 2, 1, 0, 1, 2, -1, 2, 1

Offset: 1

Witold Tatkiewicz, Oct 04 2019

Place positive integers on 2D grid starting with 1 in the top left corner and continue along the square maze as in A081344.
Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around it.
n is replaced by a(n).
This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.
a(n) < 5.
Set of n such that a(n) = 4 is unbounded (conjectured).

			Consider positive integers distributed onto the plane along increasing square array:
   1  4  5 16 17 36 ...
   2  3  6 15 18 35
   9  8  7 14 19 34
  10 11 12 13 20 33
  25 24 23 22 21 32
  26 27 28 29 30 31
...
1 is not prime and in adjacent grid cells there are 2 primes: 2 and 3. Therefore a(1) = 2.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 4 primes: 2, 3, 7 and 11. Therefore a(8) = 4.
Replacing n with a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:
  2  3  *  2  *  2  *  1  .  1  *  1 ...
  *  *  3  4  2  3  1  2  1  3  2  2
  3  4  *  3  *  1  1  2  *  3  *  1
  1  *  4  *  2  2  2  *  3  *  2  2
  1  2  *  3  3  2  *  3  3  1  2  2
  .  2  3  *  2  *  4  *  2  2  2  *
  .  1  *  3  3  2  *  3  *  2  *  4
  .  2  3  *  1  1  1  3  2  4  3  *
  1  2  *  2  1  .  1  2  *  2  *  2
  1  *  2  1  .  .  1  *  3  3  1  1
  1  1  1  .  1  1  2  2  *  2  1  .
  .  1  1  1  1  *  2  2  2  *  1  1
...
In order to produce the sequence, the graph is read along its original mapping.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Minesweeper-style graph read along original mapping, replacing -1 with a "mine", and 0 with blank space.
Michael De Vlieger, Square plot of a million terms read along original mapping, with black indicating a prime and levels of gray commensurate to a(n).
Witold Tatkiewicz, link for Java program
Wikipedia, Minesweeper game

Cf. A081344 - plane mapping
Different arrangements of integers:
Cf. A326405 - antidiagonals,
Cf. A326406 - triangle maze,
Cf. A326407 - square mapping,
Cf. A326409 - Hamiltonian path,
Cf. A326410 - Ulam's spiral.

Java
```
// See Links section.
```

Block[{n = 9, s}, s = ArrayPad[Array[If[#1 < 2 #2 - 1, #2^2 + #2 - #1, (#1 - #2)^2 + #2] & @@ {#1 + #2 - 1, #2} & @@ If[Or[And[#2 < #1, EvenQ@ #1], And[#1 < #2, EvenQ@ #2]], {#1, #2}, {#2, #1}] &, {# + 1, # + 1}], 1] &@ n; Table[If[PrimeQ@ m, -1, Count[#, ?PrimeQ] &@ Union@ Map[s[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[s, m] + {##} - 2 &, {3, 3}]]], {m, n^2}]] (* _Michael De Vlieger, Oct 04 2019 *)

A326410 Minesweeper sequence of positive integers arranged on a square spiral on a 2D grid.

Original entry on oeis.org

4, -1, -1, 3, -1, 3, -1, 3, 3, 2, -1, 5, -1, 2, 2, 2, -1, 3, -1, 3, 3, 2, -1, 2, 1, 0, 2, 3, -1, 3, -1, 3, 3, 1, 2, 2, -1, 3, 3, 2, -1, 3, -1, 1, 1, 2, -1, 2, 1, 1, 1, 1, -1, 2, 3, 2, 2, 2, -1, 2, -1, 2, 2, 1, 3, 3, -1, 1, 2, 3, -1, 4, -1, 3, 2, 0, 1, 2, -1, 1, 1

Offset: 1

Witold Tatkiewicz, Oct 07 2019

Place positive integers on a 2D grid starting with 1 in the center and continue along a spiral.
Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around it.
n is replaced by a(n).
This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.
a(n) = 5 for n = 12.
Set of n such that a(n) = 4 is unbounded (conjecture).

			Consider positive integers distributed onto the plane along the square spiral:
.
  37--36--35--34--33--32--31
   |                       |
  38  17--16--15--14--13  30
   |   |               |   |
  39  18   5---4---3  12  29
   |   |   |       |   |   |
  40  19   6   1---2  11  28
   |   |   |           |   |
  41  20   7---8---9--10  27
   |   |                   |
  42  21--22--23--24--25--26
   |
  43--44--45--46--47--48--49--...
.
1 is not prime and in adjacent grid cells there are 4 primes: 2, 3, 5 and 7. Therefore a(1) = 4.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 4 primes: 2, 7 and 23. Therefore a(8) = 3.
Replacing n with a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:
  *---2---2---1---3---3---*
  |                       |
  3   *---2---2---2---*   3
  |   |               |   |
  3   3   *---3---*   5   *
  |   |   |       |   |   |
  2   *   3   4---*   *   3
  |   |   |           |   |
  *   3   *---3---3---2   2
  |   |                   |
  3   3---2---*---2---1---.
  |
  *---1---1---2---*---2---1---...
In order to produce the sequence, the graph is read along the square spiral.

Michael De Vlieger, Table of n, a(n) for n = 1..10201 (51 spiral iterations).
Michael De Vlieger, Minesweeper-style graph read along original mapping, replacing -1 with a "mine", and 0 with blank space.
Michael De Vlieger, Square plot of 10^3 spiral iterations read along original mapping, with black indicating a prime and levels of gray commensurate to a(n).
Wikipedia, Minesweeper game

Cf. A136626 - similar sequence: For every number n in Ulam's spiral the sequence gives the number of primes around it (number n excluded).
Cf. A136627 - similar sequence: For every number n in Ulam's spiral the sequence gives the number of primes around it (number n included).
Different arrangements of integers:
Cf. A326405 (antidiagonals), A326406 (triangle maze), A326407 (square mapping), A326408 (square maze), A326409 (Hamiltonian path).

A326409 Minesweeper sequence of positive integers arranged on a 2D grid along Hamiltonian path.

Original entry on oeis.org

2, -1, -1, 3, -1, 3, -1, 3, 4, 2, -1, 3, -1, 3, 3, 2, -1, 4, -1, 2, 2, 1, -1, 2, 3, 1, 1, 2, -1, 3, -1, 3, 3, 2, 3, 2, -1, 1, 2, 2, -1, 2, -1, 2, 2, 2, -1, 1, 1, 0, 1, 2, -1, 2, 3, 1, 2, 2, -1, 2, -1, 1, 1, 1, 1, 2, -1, 1, 2, 1, -1, 3, -1, 2, 2, 1, 2, 3, -1, 1

Offset: 1

Witold Tatkiewicz, Oct 07 2019

Place positive integers on a 2D grid starting with 1 in the top left corner and continue along Hamiltonian path A163361 or A163363.
Replace each prime with -1 and each nonprime by the number of primes in adjacent grid cells around it.
n is replaced by a(n).
This sequence treats prime numbers as "mines" and fills gaps according to rules of the classic Minesweeper game.
a(n) < 5.
Set of n such that a(n) = 4 is unbounded (conjectured).

			Consider positive integers distributed onto the plane along an increasing Hamiltonian path (in this case it starts downwards):
.
   1   4---5---6  59--60--61  64--...
   |   |       |   |       |   |
   2---3   8---7  58--57  62--63
           |           |
  15--14   9--10  55--56  51--50
   |   |       |   |       |   |
  16  13--12--11  54--53--52  49
   |                           |
  17--18  31--32--33--34  47--48
       |   |           |   |
  20--19  30--29  36--35  46--45
   |           |   |           |
  21  24--25  28  37  40--41  44
   |   |   |   |   |   |   |   |
  22--23  26--27  38--39  42--43
.
1 is not prime and in adjacent grid cells there are 2 primes: 2 and 3. Therefore a(1) = 2.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 3 primes: 5, 3 and 7. Therefore a(8) = 3.
Replacing n with a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:
  2   3---*---3   *---2---*   1 ...
  |   |       |   |       |   |
  *---*   3---*   2---2   1---1
          |           |
  3---3   4---2   3---1   1---.
  |   |       |   |       |   |
  2   *---3---*   2---*---2   1
  |                           |
  *---4   *---3---3---2   *---1
      |   |           |   |
  2---*   3---*   2---3   2---2
  |           |   |           |
  2   2---3   2   *   2---*   2
  |   |   |   |   |   |   |   |
  1---*   1---1   1---2   2---*
In order to produce the sequence, the graph is read along its original mapping.

Alexander Bogomolny, Plane Filling Curves: Hilbert's & Moore's, Cut the Knot.org, retrieved October 2019.
Eric Weisstein's World of Mathematics, Hilbert Curve.
Wikipedia, Hilbert Curve.
Wikipedia, Minesweeper game.

Cf. A163361 (plane mapping), A163363 (alternative plane mapping).

Different arrangements of integers: A326405 (antidiagonals), A326406 (triangle maze), A326407 (square mapping), A326408 (square maze), A326410 (Ulam's spiral).

Block[{nn = 4, s, t, u}, s = ConstantArray[0, {2^#, 2^#}] &[nn + 1]; t = First[HilbertCurve@ # /. Line -> List] &[nn + 1] &[nn + 1]; s = ArrayPad[ReplacePart[s, Array[{1, 1} + t[[#]] -> # &, 2^(2 (nn + 1))]], {{1, 0}, {1, 0}}]; u = Table[If[PrimeQ@ m, -1, Count[#, _?PrimeQ] &@ Union@ Map[s[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[s, m] + {##} - 2 &, {3, 3}]]], {m, (2^nn)^2}]]

A327893 Minesweeper sequence of positive integers arranged in a hexagonal spiral.

Original entry on oeis.org

4, -1, -1, 3, -1, 3, -1, 3, 2, 4, -1, 3, -1, 3, 2, 2, -1, 3, -1, 2, 0, 2, -1, 3, 1, 2, 2, 2, -1, 2, -1, 1, 1, 1, 2, 3, -1, 2, 0, 1, -1, 4, -1, 1, 1, 2, -1, 1, 0, 1, 2, 3, -1, 3, 1, 0, 0, 1, -1, 4, -1, 1, 0, 0, 1, 3, -1, 2, 2, 2, -1, 2, -1, 3, 1, 1, 0, 1, -1, 3

Offset: 1

Michael De Vlieger, Oct 09 2019

Place positive integers on a 2D grid starting with 1 in the center and continue along a hexagonal spiral. Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around it. n is replaced by a(n). This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.

The largest term in the sequence is 4 since 1 is surrounded by 3 odd numbers {3, 5, 7} and the only even prime. Additionally, the pattern of odd and even numbers appears in alternating rows oriented in a triangular symmetry such that no other number has more than four odd numbers. (This courtesy of Witold Tatkiewicz.)

			Consider a spiral grid drawn counterclockwise with the largest number k = A003219(n) = 3*n*(n+1)+1 in "shell" n, and each shell has A008458(n) elements:
          28--27--26--25
          /             \
        29  13--12--11  24
        /   /         \   \
      30  14   4---3  10  23
      /   /   /     \   \   \
    31  15   5   1---2   9  22
      \   \   \         /   /
      32  16   6---7---8  21
        \   \             /
        33  17--18--19--20  ...
          \                /
          34--35--36--37--38
1 is not prime and in the 6 adjacent cells 2 through 7 inclusive, we have 4 primes, therefore a(1) = 4.
2 is prime therefore a(2) = -1.
4 is not prime and in the 6 adjacent cells {1, 3, 12, 13, 14, 5} there are 4 primes, therefore a(4) = 4, etc.
Replacing n with a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:
         2---2---2---1
        /             \
       *   *---3---*   3
      /   /         \   \
     2   3   3---*   4   *
    /   /   /     \   \   \
   *   2   *   4---*   2   2
    \   \   \         /   /
     1   3   3---*---3   .
      \   \             /
       1   *---3---*---2  ...
        \                 /
         1---2---3---*---2

Michael De Vlieger, Table of n, a(n) for n = 1..10621 (60 concentric hexagons).
Michael De Vlieger, Minesweeper style hexagonal plot of 1261 terms, replacing -1 with n in a black circle, and 0 represented by blank space.
Michael De Vlieger, Hexagonal plot of 30301 terms, (100 concentric hexagons), color coded.
Michael De Vlieger, Hexagonal plot of 120601 terms, (200 concentric hexagons), color coded.
Michael De Vlieger, Plot of 469 terms, with 12 concentric hexagons smoothed into concentric rings, color coded.
Michael De Vlieger, Plot of 120601 terms, with 200 concentric hexagons smoothed into concentric rings, color coded.

Cf. A003215, A008458, A326405, A326406, A326407, A326408, A326409, A326410.

Block[{n = 6, m, s, t, u}, m = n + 1; s = Array[3 #1 (#1 - 1) + 1 + #2 #1 + #3 & @@ {#3, #4, Which[Mod[#4, 3] == 0, Abs[#1], Mod[#4, 3] == 1, Abs[#2], True, Abs[#2] - Abs[#1]]} & @@ {#1, #2, If[UnsameQ @@ Sign[{#1, #2}], Abs[#1] + Abs[#2], Max[Abs[{#1, #2}]]], Which[And[#1 > 0, #2 <= 0], 0, And[#1 >= #2, #1 + #2 > 0], 1, And[#2 > #1, #1 >= 0], 2, And[#1 < 0, #2 >= 0], 3, And[#1 <= #2, #1 + #2 < 0], 4, And[#1 > #2, #1 + #2 <= 0], 5, True, 0]} & @@ {#2 - m - 1, m - #1 + 1} &, {#, #}] &[2 m + 1]; t = s /. k_ /; k > 3 n (n + 1) + 1 :> -k; Table[If[PrimeQ@ m, -1, Count[#, _?PrimeQ] &@ Union@ Map[t[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[t, m] + {##} - 2 & @@ {#1, #2 + Boole[#1 == #2 == 2] + Boole[#1 == 1]} &, {3, 2}]]], {m, 3 n (n - 1) + 1}]]

cp's OEIS Frontend

A326411 Triangle T(n,k) read by rows: T(n,k) = the number of ways of seating n people around a table for the second time so that k pairs are maintained. Reflected and rotated sequences are counted as one.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Java

Maple

Mathematica

PARI

Formula

A326405 Minesweeper sequence of positive integers arranged on a 2D grid along ascending antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Java

Mathematica

A326406 Minesweeper sequence of positive integers arranged on a 2D grid along a triangular maze.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Java

Mathematica

A326408 Minesweeper sequence of positive integers arranged on a 2D grid along a square maze.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Java

Mathematica

A326410 Minesweeper sequence of positive integers arranged on a square spiral on a 2D grid.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A326409 Minesweeper sequence of positive integers arranged on a 2D grid along Hamiltonian path.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A327893 Minesweeper sequence of positive integers arranged in a hexagonal spiral.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica