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Numbers that are "unsafe" when playing the game Dollar Nim, which is a Nim game where users can remove 1, 5, 10, or 25 cents from an initial pile of money. The most common version of the game is played with an initial amount of $1, hence the name.
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%I #20 Sep 08 2022 08:45:57

%S 0,2,4,6,8,15,17,19,21,23,30,32,34,36,38,45,47,49,51,53,60,62,64,66,

%T 68,75,77,79,81,83,90,92,94,96,98,105,107,109,111,113,120,122,124,126,

%U 128,135,137,139,141,143,150,152,154,156,158,165

%N Numbers that are "unsafe" when playing the game Dollar Nim, which is a Nim game where users can remove 1, 5, 10, or 25 cents from an initial pile of money. The most common version of the game is played with an initial amount of $1, hence the name.

%H Vincenzo Librandi, <a href="/A192333/b192333.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).

%F a(n) = 5*floor((n-1)/5) + 2*n - 2.

%F G.f.: x^2*(2 + 2*x + 2*x^2 + 2*x^3 + 7*x^4)/((x^4 + x^3 + x^2 + x + 1)*(x-1)^2). - _R. J. Mathar_, Jun 28 2011

%e For n = 2, a pile with 2 cents remaining is unsafe because the player would only be able to remove a penny, leaving the opponent with 1 cent and a guaranteed win.

%o (Magma) [5*Floor((n-1)/5) + 2*n - 2: n in [1..60]]; // _Vincenzo Librandi_, Jul 04 2011

%K nonn,easy

%O 1,2

%A _Patrick Vennebush_, Jun 28 2011