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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. 6
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, 63, 69, 75, 81, 87, 93, 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is the finite difference of A236305.

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.

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

LINKS

Table of n, a(n) for n=0..60.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 6

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) = 6*c-3.

EXAMPLE

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.

From Omar E. Pol, Feb 26 2015: (Start)

Also written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:

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, 63, 69, 75, 81, 87, 93;

...

Observation: the first six terms of the right border coincide with the first six terms of A068156.

(End)

From Omar E. Pol, Apr 20 2015: (Start)

An illustration of initial terms in the fourth quadrant of the square grid:

---------------------------------------------------------------------------

n   a(n)             Compact diagram

---------------------------------------------------------------------------

.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

0    1      |_| |_  |_ _ _  |_ _ _ _ _ _ _  |

1    3      |_ _| | |_ _  | |_ _ _ _ _ _  | |

2    3      | |_ _| |_  | | |_ _ _ _ _  | | |

3    9      |_ _ _ _| | | | |_ _ _ _  | | | |

4    3      | | | |_ _| | | |_ _ _  | | | | |

5    9      | | |_ _ _ _| | |_ _  | | | | | |

6   15      | |_ _ _ _ _ _| |_  | | | | | | |

7   21      |_ _ _ _ _ _ _ _| | | | | | | | |

8    3      | | | | | | | |_ _| | | | | | | |

9    9      | | | | | | |_ _ _ _| | | | | | |

10  15      | | | | | |_ _ _ _ _ _| | | | | |

11  21      | | | | |_ _ _ _ _ _ _ _| | | | |

12  27      | | | |_ _ _ _ _ _ _ _ _ _| | | |

13  33      | | |_ _ _ _ _ _ _ _ _ _ _ _| | |

14  39      | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |

15  45      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|

.

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.

(End)

MATHEMATICA

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

CROSSREFS

Cf. A011782, A068156, A236305 (partial sums), A241718 (4 piles), A241731 (5 piles).

Sequence in context: A151710 A160121 A048883 * A217883 A036553 A166466

Adjacent sequences:  A241714 A241715 A241716 * A241718 A241719 A241720

KEYWORD

nonn

AUTHOR

Tanya Khovanova and Joshua Xiong, Apr 27 2014

STATUS

approved

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Last modified February 19 09:11 EST 2018. Contains 299330 sequences. (Running on oeis4.)