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.
Original entry on oeis.org
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
Offset: 0
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)
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Table[Length[Select[Flatten[Table[{n, k, BitXor[n, k]}, {n, 0, a}, {k, 0, a}], 1], Max[#] == a &]], {a, 0, 100}]
A256534
Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).
Original entry on oeis.org
0, 4, 16, 28, 64, 76, 112, 172, 256, 268, 304, 364, 448, 556, 688, 844, 1024, 1036, 1072, 1132, 1216, 1324, 1456, 1612, 1792, 1996, 2224, 2476, 2752, 3052, 3376, 3724, 4096, 4108, 4144, 4204, 4288, 4396, 4528, 4684, 4864, 5068, 5296, 5548, 5824, 6124, 6448, 6796, 7168, 7564, 7984, 8428, 8896, 9388, 9904, 10444, 11008
Offset: 0
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
4;
16;
28, 64;
76, 112, 172, 256;
268, 304, 364, 448, 556, 688, 844, 1024;
...
Right border gives the elements of A000302 greater than 1.
This triangle T(n,k) shares with the triangle A160410 the terms of the column k, if k is a power of 2, for example, both triangles share the following terms: 4, 16, 28, 64, 76, 112, 256, 268, 304, 448, 1024, etc.
.
Illustration of initial terms, for n = 1..10:
. _ _ _ _ _ _ _ _
. | _ _ | | _ _ |
. | | _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_ | |
. | |_| _ _ _ _ _ _ _ _ _ _ _ _ |_| |
. |_ _| | _ _ _ _ | | _ _ _ _ | |_ _|
. | | | _ _ | | | | _ _ | | |
. | | | | _|_|_|_ _|_|_|_ | | | |
. | | | |_| _ _ _ _ |_| | | |
. | | |_ _| | _|_ _|_ | |_ _| | |
. | |_ _ _| |_| _ _ |_| |_ _ _| |
. | | | | | | | |
. | _ _ _| _| |_ _| |_ |_ _ _ |
. | | _ _| | |_ _ _ _| | |_ _ | |
. | | | _| |_ _| |_ _| |_ | | |
. | | | | |_ _ _ _ _ _ _ _| | | | |
. | | | |_ _| | | | | |_ _| | | |
. _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
. | _| |_ _ _ _ _ _| |_ _ _ _ _ _| |_ |
. | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
. | |_ _| | | |_ _| |
. |_ _ _ _| |_ _ _ _|
.
After 10 generations there are 304 ON cells, so a(10) = 304.
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{0}~Join~Flatten@ Table[4^i + 3 (2 j)^2, {i, 6}, {j, 0, 2^(i - 1) - 1}] (* Michael De Vlieger, Nov 03 2022 *)
A241522
The number of P-positions in the game of Nim with up to 4 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.
Original entry on oeis.org
1, 8, 21, 64, 89, 168, 301, 512, 561, 712, 965, 1344, 1801, 2408, 3165, 4096, 4193, 4488, 4981, 5696, 6585, 7720, 9101, 10752, 12433, 14408, 16677, 19264, 22121, 25320, 28861, 32768, 32961, 33544, 34517, 35904, 37657, 39848, 42477, 45568, 48881, 52680, 56965, 61760, 67017
Offset: 0
If the largest number is 1, then there should be an even number of piles of size 1. Thus, a(1)=8.
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Table[Length[Select[Flatten[Table[{n, k, j, BitXor[n, k, j]}, {n, 0, a}, {k, 0, a}, {j, 0, a}], 2], #[[4]] <= a &]], {a, 0, 50}]
A241523
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.
Original entry on oeis.org
1, 16, 61, 256, 421, 976, 2101, 4096, 4741, 6736, 10261, 15616, 23221, 33616, 47461, 65536, 68101, 75856, 88981, 107776, 132661, 164176, 202981, 249856, 305701, 371536, 448501, 537856, 640981, 759376, 894661, 1048576, 1058821, 1089616
Offset: 0
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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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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