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%I #41 Feb 28 2018 15:05:35
%S 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 63,69,75,81,87,93,3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93,99,
%U 105,111,117,123,129,135,141,147,153,159,165,171
%N 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 This is the finite difference of A236305.
%C 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 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 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 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 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 From _Omar E. Pol_, Feb 26 2015: (Start)
%e Also written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
%e 1;
%e 3;
%e 3, 9;
%e 3, 9, 15, 21;
%e 3, 9, 15, 21, 27, 33, 39, 45;
%e 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93;
%e ...
%e Observation: the first six terms of the right border coincide with the first six terms of A068156.
%e (End)
%e From _Omar E. Pol_, Apr 20 2015: (Start)
%e An illustration of initial terms in the fourth quadrant of the square grid:
%e ---------------------------------------------------------------------------
%e n a(n) Compact diagram
%e ---------------------------------------------------------------------------
%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e 0 1 |_| |_ |_ _ _ |_ _ _ _ _ _ _ |
%e 1 3 |_ _| | |_ _ | |_ _ _ _ _ _ | |
%e 2 3 | |_ _| |_ | | |_ _ _ _ _ | | |
%e 3 9 |_ _ _ _| | | | |_ _ _ _ | | | |
%e 4 3 | | | |_ _| | | |_ _ _ | | | | |
%e 5 9 | | |_ _ _ _| | |_ _ | | | | | |
%e 6 15 | |_ _ _ _ _ _| |_ | | | | | | |
%e 7 21 |_ _ _ _ _ _ _ _| | | | | | | | |
%e 8 3 | | | | | | | |_ _| | | | | | | |
%e 9 9 | | | | | | |_ _ _ _| | | | | | |
%e 10 15 | | | | | |_ _ _ _ _ _| | | | | |
%e 11 21 | | | | |_ _ _ _ _ _ _ _| | | | |
%e 12 27 | | | |_ _ _ _ _ _ _ _ _ _| | | |
%e 13 33 | | |_ _ _ _ _ _ _ _ _ _ _ _| | |
%e 14 39 | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e 15 45 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e .
%e 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 (End)
%t Table[Length[Select[Flatten[Table[{n, k, BitXor[n, k]}, {n, 0, a}, {k, 0, a}], 1], Max[#] == a &]], {a, 0, 100}]
%Y Cf. A011782, A068156, A236305 (partial sums), A241718 (4 piles), A241731 (5 piles).
%K nonn
%O 0,2
%A _Tanya Khovanova_ and _Joshua Xiong_, Apr 27 2014
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