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
KEYWORD
nonn
AUTHOR
Tanya Khovanova and Joshua Xiong, Apr 27 2014
STATUS
approved