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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 T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 6 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) = 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 May 27 13:39 EDT 2018. Contains 304693 sequences. (Running on oeis4.)