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The number of P-positions in the game of Nim with up to 5 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.

5

`%I #19 Feb 28 2018 15:06:01
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`%S 1,16,61,256,421,976,2101,4096,4741,6736,10261,15616,23221,33616,
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`%T 47461,65536,68101,75856,88981,107776,132661,164176,202981,249856,
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`%U 305701,371536,448501,537856,640981,759376,894661,1048576,1058821,1089616
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`%N The number of P-positions in the game of Nim with up to 5 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.
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`%C P-positions in the game of Nim are tuples of numbers with a Nim-Sum equal to zero. (0,1,1,0,0) is considered different from (1,0,1,0,0).
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`%C a(2^n-1) = 2^(4n).
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`%H T. Khovanova and J. Xiong, <a href="http://arxiv.org/abs/1405.5942">Nim Fractals</a>, arXiv:1405.594291 [math.CO] (2014), p. 9 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Khovanova/khova6.html">J. Int. Seq. 17 (2014) # 14.7.8</a>.
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`%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) = 2^(4*b) + 10*2^(2*b)*c^2 + 5*c^4.
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`%e If the largest number is not more than 1, then there should be an even number of piles of size 1. We can choose the first four piles to be either 0 or 1, then the last pile is uniquely defined. Thus, a(1)=16.
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`%t Table[Length[Select[Flatten[Table[{n, k, j, i, BitXor[n, k, j, i]}, {n, 0, a}, {k, 0, a}, {j, 0, a}, {i, 0, a}], 3], #[[5]] <= a &]], {a, 0, 35}]
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`%Y Cf. A236305 (3 piles), A241522 (4 piles).
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`%Y Cf. A241731 (first differences).
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`%K nonn
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`%O 0,2
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`%A _Tanya Khovanova_ and _Joshua Xiong_, Apr 24 2014
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