A346637 -id:A346637 - cp's OEIS frontend

A361517 The value of n for which the two-player impartial {0,1}-Toggle game on a generalized Petersen graph GP(n,2) with a (1,0)-weight assignment is a next-player winning game.

3, 4, 5, 11, 17, 27, 35, 37, 49, 59, 69, 81, 91, 103, 115, 123, 135, 137, 167, 175, 189, 199, 207, 287, 295, 307, 361, 1051, 2507, 2757, 2917, 3057, 3081, 7255, 7361, 7871, 16173

Offset: 3

Eugene Fiorini, Maxwell Fogler, Katherine Levandosky, Bryan Lu, Jacob K. Porter and Andrew Woldar, Mar 14 2023

The two-player impartial {0,1}-Toggle game is played on a simple connected graph G where each vertex is assigned an initial weight of 0 or 1.
A Toggle move consists of selecting a vertex v and switching its weight as well as the weights of each of its neighbors. This move is only legal provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.
In the special case G=GP(n,2), a (1,0)-weight assignment is one in which each vertex of the outer polygon is assigned weight 1 and each vertex of the inner polygon(s) is assigned weight 0.

			For n = 3, the {0,1}-Toggle game on GP(3,2) with a (1,0)-weight assignment is a next-player winning game.
For n = 5, the {0,1}-Toggle game on GP(5,2) with a (1,0)-weight assignment is a next-player winning game.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Eugene Fiorini, Maxwell Fogler, Katherine Levandosky, Bryan Lu, Jacob Porter, and Andrew Woldar, On the Nature and Complexity of an Impartial Two-Player Variant of the Game Lights-Out, arXiv:2411.08247 [math.CO], 2024. See p. 17.
E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs, Journal of Integer Sequences, 24(6), 2021.
Katherine Levandosky, CGSuite Program.

Cf. A071426, A340631, A346197, A346401, A346637.

CGSuite
```
# See Levandosky link
```

A346636 a(n) is the number of quadruples (a_1, a_2, a_3, a_4) having all terms in {1,...,n} such that there exists a quadrilateral with these side lengths.

Original entry on oeis.org

0, 1, 16, 77, 236, 565, 1156, 2121, 3592, 5721, 8680, 12661, 17876, 24557, 32956, 43345, 56016, 71281, 89472, 110941, 136060, 165221, 198836, 237337, 281176, 330825, 386776, 449541, 519652, 597661, 684140, 779681, 884896, 1000417, 1126896, 1265005, 1415436

Offset: 0

Giovanni Corbelli, Jul 26 2021

nonn

The existence of such a four-sided polygon implies that every element of the quadruple is less than the sum of the other elements.

Giovanni Corbelli, Visual Basic routine for generating number of four-sided polygons
Giovanni Corbelli Proof of the formula: Number of k-tuples with elements in {1,2,…,N} corresponding to k-sided polygons
Sean A. Irvine, Java program (github)

Cf. A006003, A346637, A346638.

Formula: a(n) = n^4 - 4*binomial(n+1,4) = n^4 - (n+1)*binomial(n,3).

General formula for k-tuples: a_k(n) = n^k - k*binomial(n+1,k) = n^k - (n+1)*binomial(n,k-1).

A346638 a(n) is the number of 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n} such that there exists a hexagon with these side-lengths.

Original entry on oeis.org

0, 1, 64, 729, 4096, 15619, 46614, 117481, 261640, 530181, 997228, 1766017, 2975688, 4808791, 7499506, 11342577, 16702960, 24026185, 33849432, 46813321, 63674416, 85318443, 112774222, 147228313, 190040376, 242759245, 307139716, 385160049, 479040184, 591260671

Offset: 0

Giovanni Corbelli, Jul 26 2021

The existence of such a six-sided polygon implies that every element of the sextuple is less than the sum of the other elements.

Giovanni Corbelli, Visual Basic routine for generating number of six-sided polygons
Giovanni Corbelli Proof of the formula: Number of k-tuples with elements in {1,2,...,N} corresponding to k-sided polygons
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006003, A346636, A346637.

CoefficientList[Series[x (1+57x+302x^2+302x^3+51x^4+x^5)/(1-x)^7,{x,0,40}],x] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,64,729,4096,15619,46614},40] (* Harvey P. Dale, Oct 30 2024 *)

a(n) = n^6 - 6*binomial(n+1,6) = n^6 - (n+1)*binomial(n,5).
General formula for k-tuples: a_k(n) = n^k - k*binomial(n+1,k) = n^k - (n+1)*binomial(n,k-1).
G.f.: x*(1 + 57*x + 302*x^2 + 302*x^3 + 51*x^4 + x^5)/(1 - x)^7. - Stefano Spezia, Sep 27 2021

A374810 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+2) = v(1)v(2)...v(k+2) with a (1^k,0,1)-weight assignment is a second-player winning game.

Original entry on oeis.org

1, 6, 7, 12, 13, 18, 23, 24, 38, 39, 44, 45, 50, 51, 56, 62, 77, 115, 121, 153, 312, 333, 350, 427, 553, 554, 579

Offset: 1

Patrick G. Cesarz, Matthew Cohen, Eugene Fiorini, Joshua Lowrance, Emily Riley, Angel Pedro Torres and Andrew Woldar, Jul 20 2024

The two-player impartial {0,1}-Toggle game is played on a simple connected graph G where each vertex is assigned an initial weight of 0 or 1.
A Toggle move consists of selecting a vertex v and switching its weight as well as the weight of each of its neighbors. This move is legal only provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.
In the special case G = P(k+2), a (1^k, 0, 1)-weight assignment is one in which vertex v(k+1) is assigned weight 0 and all remaining vertices are assigned weight 1.

			For n = 6, the {0,1}-Toggle game on P(8) with a (1,1,1,1,1,1,0,1)-weight assignment is a second-player winning game.
For n = 12, the {0,1}-Toggle game on P(14) with a (1,1,1,1,1,1,1,1,1,1,1,1,0,1)-weight assignment is a second-player winning game.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.

K. Barker, M. DeStefano, E. Fiorini, M. Gohn, J. Miller, J. Roeder, and T. W. H. Wong, Generalized Impartial Two-player Pebbling Games on K3 and C4, Journal of Integer Sequences, 27(5), 2024.
Matthew Cohen, Python
E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs, Journal of Integer Sequences, 24(6), 2021.

Cf. A071426, A340631, A346197, A346401, A346637, A361517, A374910, A374920

Python
```
# See Cohen link.
```

A374910 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+4) = v(1)v(2)...v(k+4) with a (1^k,0,1,0,1)-weight assignment is a second-player winning game.

Original entry on oeis.org

1, 25, 26, 31, 32, 37, 38, 63, 64, 69, 70, 76, 101, 102, 139, 145, 177, 189, 215, 235, 252, 253, 267, 284, 290, 305, 311, 328, 360, 668

Offset: 1

Patrick G. Cesarz, Matthew Cohen, Eugene Fiorini, Joshua Lowrance, Emily Riley, Angel Pedro Torres and Andrew Woldar, Jul 20 2024

The two-player impartial {0,1}-Toggle game is played on a simple connected graph G where each vertex is assigned an initial weight of 0 or 1.
A Toggle move consists of selecting a vertex v and switching its weight as well as the weight of each of its neighbors. This move is only legal provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.
In the special case G=P(k+4), a (1^k,0,1,0,1)-weight assignment is one in which vertices v(k+1) and v(k+3) are assigned weight 0 and all remaining vertices are assigned weight 1.
The path P(k+4m) where vertices v(k+1), v(k+3), ..., v(k+4m-1) are assigned weight 0 and all remaining vertices are assigned weight 1 will have the same Grundy numbers as G.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.

K. Barker, M. DeStefano, E. Fiorini, M. Gohn, J. Miller, J. Roeder, and T. W. H. Wong, Generalized Impartial Two-player Pebbling Games on K3 and C4, Journal of Integer Sequences, 27(5), 2024.
Matthew Cohen, Python
E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs, Journal of Integer Sequences, 24(6), 2021.

Cf. A071426, A340631, A346197, A346401, A346637, A361517, A374810, A374920.

Python
```
# See Cohen link.
```

A374920 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+6) = v(1)v(2)...v(k+6) with a (1^k,0,1,0,1,0,1)-weight assignment is a second-player winning game.

Original entry on oeis.org

1, 6, 7, 12, 13, 18, 23, 24, 39, 44, 45, 50, 51, 57, 62, 77, 115, 281, 319, 350, 389

Offset: 1

Patrick G. Cesarz, Matthew Cohen, Eugene Fiorini, Joshua Lowrance, Emily Riley, Angel Pedro Torres and Andrew Woldar, Jul 20 2024

The two-player impartial {0,1}-Toggle game is played on a simple connected graph G where each vertex is assigned an initial weight of 0 or 1.
A Toggle move consists of selecting a vertex v and switching its weight as well as the weight of each of its neighbors. This move is only legal provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.
In the special case G=P(k+6), a (1^k,0,1,0,1,0,1)-weight assignment is one in which vertices v(k+1), v(k+3), and v(k+5) are assigned weight 0 and all remaining vertices are assigned weight 1.
The path P(k+4m+2) where vertices v(k+1), v(k+3), ..., v(k+4m+1) are assigned weight 0 and all remaining vertices are assigned weight 1 will have the same Grundy numbers as G.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.

K. Barker, M. DeStefano, E. Fiorini, M. Gohn, J. Miller, J. Roeder, and T. W. H. Wong, Generalized Impartial Two-player Pebbling Games on K3 and C4, Journal of Integer Sequences, 27(5), 2024.
Matthew Cohen, Python
E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs, Journal of Integer Sequences, 24(6), 2021.

Cf. A071426, A340631, A346197, A346401, A346637, A361517, A374810, A374910

Python
```
# See Cohen link.
```

cp's OEIS Frontend

A361517 The value of n for which the two-player impartial {0,1}-Toggle game on a generalized Petersen graph GP(n,2) with a (1,0)-weight assignment is a next-player winning game.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

CGSuite

A346636 a(n) is the number of quadruples (a_1, a_2, a_3, a_4) having all terms in {1,...,n} such that there exists a quadrilateral with these side lengths.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A346638 a(n) is the number of 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n} such that there exists a hexagon with these side-lengths.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A374810 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+2) = v(1)v(2)...v(k+2) with a (1^k,0,1)-weight assignment is a second-player winning game.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Python

A374910 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+4) = v(1)v(2)...v(k+4) with a (1^k,0,1,0,1)-weight assignment is a second-player winning game.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Python

A374920 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+6) = v(1)v(2)...v(k+6) with a (1^k,0,1,0,1,0,1)-weight assignment is a second-player winning game.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Python