A014587 -id:A014587 - cp's OEIS frontend

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

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

A355556 a(n) is the smallest position in the subtract-a-factorial game for which the value of the Sprague-Grundy function (or nim-value) is n.

Original entry on oeis.org

0, 1, 2, 6, 5050, 5056, 5064, 40520, 40696, 630373, 40348521, 483383076, 6302798387

Offset: 0

Pontus von Brömssen, Jul 09 2022

			a(3) = 6, because the smallest k for which A014587(k) = 3 is k = 6.

Rémy Sigrist, C program

Cf. A014587, A019307, A297963, A355557.

C
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See Links section.
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a(11)-a(12) from Rémy Sigrist, Jul 09 2022

A308898 Fixed point of the morphism 0 -> 01, 1 -> 2, 2 -> 3, 3 -> 012.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 24 2019

nonn

This is superficially similar to A014587, although the latter sequence is not well understood. - N. J. A. Sloane, Jul 28 2019

Rémy Sigrist, Table of n, a(n) for n = 0..15126
M. Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], June 2019.

Cf. A014587. See A309198 for another version.

Nest[Flatten[ReplaceAll[#,{0->{0,1},1->2,2->3,3->{0,1,2}}]]&,{0},10] (* Paolo Xausa, Nov 09 2023 *)

M = [[1,2], [3], [4], [1,2,3]]; s = [1]; for (k=1, 100, while (k>#s, s=concat(apply(d -> M[d],s))); print1 (s[k]-1 ", ")); \\ Rémy Sigrist, Jul 25 2019

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

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A355556 a(n) is the smallest position in the subtract-a-factorial game for which the value of the Sprague-Grundy function (or nim-value) is n.

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A308898 Fixed point of the morphism 0 -> 01, 1 -> 2, 2 -> 3, 3 -> 012.

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