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A241717 The number of P-positions in the game of Nim with up to 3 piles, allowing for piles of zero, such that the number of objects in the largest pile is n.

%I A241717 #41 Feb 28 2018 15:05:35
%S A241717 1,3,3,9,3,9,15,21,3,9,15,21,27,33,39,45,3,9,15,21,27,33,39,45,51,57,
%T A241717 63,69,75,81,87,93,3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93,99,
%U A241717 105,111,117,123,129,135,141,147,153,159,165,171
%N A241717 The number of P-positions in the game of Nim with up to 3 piles, allowing for piles of zero, such that the number of objects in the largest pile is n.
%C A241717 This is the finite difference of A236305.
%C A241717 Starting from index 1 all elements are divisible by 3, and can be grouped into sets of size 2^k of an arithmetic progression 6n-3.
%C A241717 It appears that the sum of all terms of the first n rows of triangle gives A000302(n-1), see Example section. - _Omar E. Pol_, May 01 2015
%H A241717 T. Khovanova and J. Xiong, <a href="http://arxiv.org/abs/1405.5942">Nim Fractals</a>, arXiv:1405.594291 [math.CO] (2014), p. 6 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Khovanova/khova6.html">J. Int. Seq. 17 (2014) # 14.7.8</a>.
%F A241717 If b = floor(log_2(n)) is the number of digits in the binary representation of n and c = n + 1 - 2^b, then a(n) = 6*c-3.
%e A241717 If the largest number is 1, then there should be exactly two piles of size 1 and one empty pile. There are 3 ways to permute this configuration, so a(1)=3.
%e A241717 From _Omar E. Pol_, Feb 26 2015: (Start)
%e A241717 Also written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
%e A241717 1;
%e A241717 3;
%e A241717 3, 9;
%e A241717 3, 9, 15, 21;
%e A241717 3, 9, 15, 21, 27, 33, 39, 45;
%e A241717 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93;
%e A241717 ...
%e A241717 Observation: the first six terms of the right border coincide with the first six terms of A068156.
%e A241717 (End)
%e A241717 From _Omar E. Pol_, Apr 20 2015: (Start)
%e A241717 An illustration of initial terms in the fourth quadrant of the square grid:
%e A241717 ---------------------------------------------------------------------------
%e A241717 n   a(n)             Compact diagram
%e A241717 ---------------------------------------------------------------------------
%e A241717 .            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A241717 0    1      |_| |_  |_ _ _  |_ _ _ _ _ _ _  |
%e A241717 1    3      |_ _| | |_ _  | |_ _ _ _ _ _  | |
%e A241717 2    3      | |_ _| |_  | | |_ _ _ _ _  | | |
%e A241717 3    9      |_ _ _ _| | | | |_ _ _ _  | | | |
%e A241717 4    3      | | | |_ _| | | |_ _ _  | | | | |
%e A241717 5    9      | | |_ _ _ _| | |_ _  | | | | | |
%e A241717 6   15      | |_ _ _ _ _ _| |_  | | | | | | |
%e A241717 7   21      |_ _ _ _ _ _ _ _| | | | | | | | |
%e A241717 8    3      | | | | | | | |_ _| | | | | | | |
%e A241717 9    9      | | | | | | |_ _ _ _| | | | | | |
%e A241717 10  15      | | | | | |_ _ _ _ _ _| | | | | |
%e A241717 11  21      | | | | |_ _ _ _ _ _ _ _| | | | |
%e A241717 12  27      | | | |_ _ _ _ _ _ _ _ _ _| | | |
%e A241717 13  33      | | |_ _ _ _ _ _ _ _ _ _ _ _| | |
%e A241717 14  39      | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e A241717 15  45      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e A241717 .
%e A241717 It appears that a(n) is also the number of cells in the n-th region of the diagram, and A236305(n) is also the total number of cells after n-th stage.
%e A241717 (End)
%t A241717 Table[Length[Select[Flatten[Table[{n, k, BitXor[n, k]}, {n, 0, a}, {k, 0, a}], 1], Max[#] == a &]], {a, 0, 100}]
%Y A241717 Cf. A011782, A068156, A236305 (partial sums), A241718 (4 piles), A241731 (5 piles).
%K A241717 nonn
%O A241717 0,2
%A A241717 _Tanya Khovanova_ and _Joshua Xiong_, Apr 27 2014