

A236305


The number of Ppositions in the game of Nim with up to 3 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.


4



1, 4, 7, 16, 19, 28, 43, 64, 67, 76, 91, 112, 139, 172, 211, 256, 259, 268, 283, 304, 331, 364, 403, 448, 499, 556, 619, 688, 763, 844, 931, 1024, 1027, 1036, 1051, 1072, 1099, 1132, 1171, 1216, 1267, 1324, 1387, 1456, 1531, 1612, 1699
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OFFSET

0,2


COMMENTS

Ppositions in the game of Nim are tuples of numbers with a NimSum equal to zero.
(0,1,1) is considered different from (1,0,1) and (1,1,0).
a(2^n1) = 2^(2*n).
Partial sums of A241717.
This sequence seems to be A256534(n+1)/4.  Thomas Baruchel, May 15 2018


LINKS

Table of n, a(n) for n=0..46.
T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 7 and J. Int. Seq. 17 (2014) # 14.7.8.


FORMULA

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^(2*b) + 3*c^2.
a(n) = 4^floor(log(n)/log(2)) + 3*(n mod 2^floor(log(n)/log(2)))^2 (conjectured).  Thomas Baruchel, May 15 2018


EXAMPLE

If the largest number is 1, then there should be an even number of piles of size 1. Thus, a(1)=4.


MATHEMATICA

Table[Length[Select[Flatten[Table[{n, k, BitXor[n, k]}, {n, 0, a}, {k, 0, a}], 1], #[[3]] <= a &]], {a, 0, 100}]


CROSSREFS

Cf. A241522 (4 piles), A241523 (5 piles).
Cf. A241717 (first differences).
Sequence in context: A160715 A160120 A130665 * A212062 A256926 A101534
Adjacent sequences: A236302 A236303 A236304 * A236306 A236307 A236308


KEYWORD

nonn


AUTHOR

Tanya Khovanova and Joshua Xiong, Apr 21 2014


STATUS

approved



