Achim Flammenkamp - cp's OEIS frontend

A046874 Row/column pre-periods of Sprague-Grundy values of Wythoff's Game.

0, 0, 0, 8, 9, 27, 37, 92, 102, 127, 224, 277, 347, 382, 613, 693, 771, 865, 919, 1032, 1165, 1252, 1293, 1373, 1732, 2208, 2314, 2608, 2889, 3221, 3890, 4422, 4419, 5614, 6065, 6837, 7116, 8041, 8356, 9208, 10333, 10799, 12518, 13849, 14190, 15844

Offset: 0

Achim Flammenkamp

nonn

A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).

Cf. A046875, A046876.

A046875 Row/column periods of Sprague-Grundy values of Wythoff's Game.

Original entry on oeis.org

1, 3, 3, 6, 12, 24, 12, 24, 24, 24, 24, 48, 48, 96, 96, 96, 192, 192, 384, 384, 384, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 1536, 1536, 1536, 1536, 1536, 1536, 1536, 1536, 1536, 1536

Offset: 0

Achim Flammenkamp

nonn

A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).

Cf. A046874, A046876.

A039740 Lattice animals in the cubic diamond lattice (4 nearest neighbors), even-vertex-connected cubes.

Original entry on oeis.org

1, 4, 12, 44, 182, 796, 3612, 16786, 79426, 381250, 1851832, 9086022, 44970688, 224279188, 1126038472, 5687027204, 28873452032, 147281732692, 754443210294

Offset: 1

Achim Flammenkamp

nonn

A046876 Length of runs in the sequence of row/column periods of Sprague-Grundy values of Wythoff's Game (A046875).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 4, 2, 3, 2, 3, 23, 24, 3, 20, 18, 84, 25, 43

Offset: 1

Achim Flammenkamp

a(20) >= 12. - Sean A. Irvine, May 02 2021

A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).

Cf. A046874, A046875, A004481.

Entry revised by Sean A. Irvine, May 02 2021

A014586 Nim-Grundy function for Take-a-Square (or Subtract-a-Square) game.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 2, 3, 4, 5, 3, 2, 3, 4, 0, 1, 2, 3, 2, 0, 1, 2, 3, 2, 0, 1, 2, 3, 2, 3, 4, 5, 0, 1, 3, 4, 5, 0, 1, 3, 4, 5, 0, 1, 3, 0, 1, 0, 1, 2, 4, 3, 0, 1, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 0, 1, 6, 3, 2, 4, 2, 6, 4, 5, 0, 1, 6, 4

Offset: 0

Achim Flammenkamp

nonn

Concerning the January 1997 dissertation of Achim Flammenkamp, his home page (currently http://wwwhomes.uni-bielefeld.de/cgi-bin/cgiwrap/achim/index.cgi) has the link shown below, and a comment that a book was published in July 1997 by Hans-Jacobs-Verlag, Lage, Germany with the title Lange Perioden in Subtraktions-Spielen (ISBN 3-932136-10-1). This is an enlarged study (more than 200 pages) of his dissertation. - N. J. A. Sloane, Jul 25 2019

R. K. Guy, Unsolved Problems in Number Theory, E26.
W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 12th Edition.

Eric M. Schmidt, Table of n, a(n) for n = 0..10000 (corrected by _Eric M. Schmidt_, Apr 23 2019)
David Eppstein, Faster Evaluation of Subtraction Games, Proceedings of the 9th International Conference on Fun with Algorithms (FUN 2018), Leibniz International Proceedings in Informatics, arXiv:1804.06515 [cs.DS], 2018.
Achim Flammenkamp, Lange Perioden in Subtraktions-Spielen, Dissertation, Dept. Math., University of Bielefeld, Germany.
Sean A. Irvine, Java program (github)
Wikipedia, Sprague-Grundy theorem

Cf. A030193, A014587-A014589, A297963.

def A014586_list(max) :
    res = []
    for i in range(max+1) :
        moves = list({res[i-r^2] for r in range(1, isqrt(i)+1)})
        moves.sort()
        k = len(moves)
        mex = next((j for j in range(k) if moves[j] != j), k)
        res.append(mex)
    return res
A014586_list(100)
# Eric M. Schmidt, Jul 20 2013, corrected Eric M. Schmidt, Apr 23 2019

a(n) = 0 iff n belongs to A030193. - Rémy Sigrist, May 30 2019

A014588 Nim function for Take-a-Fibonacci-Game (a subtraction game).

Original entry on oeis.org

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

Offset: 0

Achim Flammenkamp

nonn

This game is also called Fibonacci nim, but there is also a different game with the same name. Its winning positions (the indexes of zeros in this sequence) are A001581 and its (much sparser) odd winning positions are A120904. - David Eppstein, Jun 14 2018

Concerning the January 1997 dissertation of Achim Flammenkamp, his home page (currently http://wwwhomes.uni-bielefeld.de/cgi-bin/cgiwrap/achim/index.cgi) has the link shown below, and a comment that a book was published in July 1997 by Hans-Jacobs-Verlag, Lage, Germany with the title Lange Perioden in Subtraktions-Spielen (ISBN 3-932136-10-1). This is an enlarged study (more than 200 pages) of his dissertation. - N. J. A. Sloane, Jul 25 2019

R. K. Guy, Unsolved Problems in Number Theory, E26.
David L. Silverman, Your Move, McGraw Hill, 1971, page 211. Reprinted by Dover Books, 1991 (mentions this game).

Eric M. Schmidt, Table of n, a(n) for n = 0..10000
Achim Flammenkamp, Lange Perioden in Subtraktions-Spielen, Dissertation, Dept. Math., University of Bielefeld, Germany.
Wikipedia, Sprague-Grundy theorem

Cf. A001581, A014586-A014589, A019307, A120904.

def A014588(max) :
    res = []
    for i in range(max+1) :
        moves = list({res[i-f] for f in fibonacci_xrange(1,i+1)})
        moves.sort()
        k = len(moves)
        mex = next((j for j in range(k) if moves[j] != j), k)
        res.append(mex)
    return res
# Eric M. Schmidt, Jul 20 2013, corrected Eric M. Schmidt, Apr 24 2019

A014587 Nim function for Take-a-Factorial-Game (a subtraction game).

Original entry on oeis.org

0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2

Offset: 0

Achim Flammenkamp

nonn

Concerning the January 1997 dissertation of Achim Flammenkamp, his home page (currently http://wwwhomes.uni-bielefeld.de/cgi-bin/cgiwrap/achim/index.cgi) has the link shown below, and a comment that a book was published in July 1997 by Hans-Jacobs-Verlag, Lage, Germany with the title Lange Perioden in Subtraktions-Spielen (ISBN 3-932136-10-1). This is an enlarged study (more than 200 pages) of his dissertation. - N. J. A. Sloane, Jul 25 2019

R. K. Guy, Unsolved Problems in Number Theory, E26.

Eric M. Schmidt, Table of n, a(n) for n = 0..10000
Achim Flammenkamp, Lange Perioden in Subtraktions-Spielen, Dissertation, Dept. Math., University of Bielefeld, Germany.

Cf. A014586-A014589, A355556.

def A014587(max) :
    res = []
    fact = [1]
    while fact[-1] <= max : fact.append(factorial(len(fact)))
    for i in range(max+1) :
        moves = list({res[i-f] for f in fact if f <= i})
        moves.sort()
        k = len(moves)
        mex = next((j for j in range(k) if moves[j] != j), k)
        res.append(mex)
    return res
# Eric M. Schmidt, Jul 20 2013, corrected Eric M. Schmidt, Apr 24 2019

Conjecture: Appears to be periodic with period of length 25 =  [0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3] starting with the initial term (there is no preamble). - Michel Dekking, Jul 26 2019
This conjecture is false, because moving from 10! = 3628800 to 0 is a legal move, and so a(3628800) cannot be zero. A similar argument shows that for no value of P is this sequence periodic with period P starting at term 0 (for a(P!) cannot be zero). - Nathan Fox, Jul 28 2019.
The first counterexample to the conjecture above is a(5050) = 4. - Pontus von Brömssen, Jul 09 2022

A038168 Number of "connected animals" formed from n tricapped truncated tetrahedra in the diamond lattice, allowing translation and rotations of the lattice.

Original entry on oeis.org

1, 1, 1, 4, 10, 39, 160, 726, 3344, 16004, 77323, 379117, 1874540

Offset: 1

Achim Flammenkamp

nonn

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.

Cf. A000162, A038119, A038169-A038174.

A038172 Number of "connected animals" formed from n rhombic dodecahedra (or edge-connected cubes) in the face-centered cubic lattice, allowing translation and rotations of the lattice.

Original entry on oeis.org

1, 1, 5, 28, 225, 2274, 24955, 286143, 3367443, 40358811, 490598186

Offset: 1

Achim Flammenkamp

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
G. L. Sicherman, Catalogue of Polyrhons
T. Sillke, Notations for polyspheres
Index entries for sequences related to f.c.c. lattice

Cf. A000162, A038119, A038168-A038174.

This extends earlier work of Torsten Sillke (TORSTEN.SILLKE(AT)LHSYSTEMS.COM). Added the 10th term.

a(11) from Joerg Arndt and Márk Péter Légrádi, Apr 30 2023

A038170 Number of "connected animals" formed from n 6-gon connected truncated octahedra (or corner-connected cubes) in the b.c.c. lattice, allowing translation and rotations of the lattice.

Original entry on oeis.org

1, 1, 3, 14, 88, 686, 5966, 54722, 516454, 4970445, 48527372, 479314799

Offset: 1

Achim Flammenkamp

nonn

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
Index entries for sequences related to b.c.c. lattice

Cf. A000162, A038119, A038168-A038174.

a(11) and a(12) from Joerg Arndt and Márk Péter Légrádi, May 02 2023

cp's OEIS Frontend

User: Achim Flammenkamp

A046874 Row/column pre-periods of Sprague-Grundy values of Wythoff's Game.

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A046875 Row/column periods of Sprague-Grundy values of Wythoff's Game.

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A039740 Lattice animals in the cubic diamond lattice (4 nearest neighbors), even-vertex-connected cubes.

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A046876 Length of runs in the sequence of row/column periods of Sprague-Grundy values of Wythoff's Game (A046875).

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A014586 Nim-Grundy function for Take-a-Square (or Subtract-a-Square) game.

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Sage

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A014588 Nim function for Take-a-Fibonacci-Game (a subtraction game).

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Sage

A014587 Nim function for Take-a-Factorial-Game (a subtraction game).

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Programs

Sage

Formula

A038168 Number of "connected animals" formed from n tricapped truncated tetrahedra in the diamond lattice, allowing translation and rotations of the lattice.

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Keywords

Links

Crossrefs

A038172 Number of "connected animals" formed from n rhombic dodecahedra (or edge-connected cubes) in the face-centered cubic lattice, allowing translation and rotations of the lattice.

Views

Author

Keywords

Links

Crossrefs

Extensions

A038170 Number of "connected animals" formed from n 6-gon connected truncated octahedra (or corner-connected cubes) in the b.c.c. lattice, allowing translation and rotations of the lattice.

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Author

Keywords

Links

Crossrefs

Extensions