A241717 -id:A241717 - cp's OEIS frontend

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 127, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137

Offset: 1

Omar E. Pol, Apr 04 2015

Row sums give A002001.
The sum of all terms of first n rows gives A000302(n-1).
The rows of triangle A256263 converge to this sequence.
Rows converge to A016969.
First 11 terms agree with A151548.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
3;
5,7;
5,11,17,15;
5,11,17,23,29,35,41,31;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,107,113,119,125,131,137,143,149,155,161,167,173,179,185,127;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n   a(n)             Compact diagram
------------------------------------------------------------------------
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1    1      |_| | | |_ _  | |_ _ _ _ _ _  | |
2    3      |_ _| | |_  | | |_ _ _ _ _  | | |
3    5      |_ _ _| | | | | |_ _ _ _  | | | |
4    7      |_ _ _ _| | | | |_ _ _  | | | | |
5    5      | | |_ _ _| | | |_ _  | | | | | |
6   11      | |_ _ _ _ _| | |_  | | | | | | |
7   17      |_ _ _ _ _ _ _| | | | | | | | | |
8   15      |_ _ _ _ _ _ _ _| | | | | | | | |
9    5      | | | | | | |_ _ _| | | | | | | |
10  11      | | | | | |_ _ _ _ _| | | | | | |
11  17      | | | | |_ _ _ _ _ _ _| | | | | |
12  23      | | | |_ _ _ _ _ _ _ _ _| | | | |
13  29      | | |_ _ _ _ _ _ _ _ _ _ _| | | |
14  35      | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
15  41      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
16  31      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
It appears that A241717 can be represented by a similar diagram.

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A000225, A000302, A002001, A011782, A016969, A141548, A241717, A256260, A256261, A256263, A256264.

Nest[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 7] (* Ivan Neretin, Feb 14 2017 *)

A236305 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 each pile does not exceed n.

Original entry on oeis.org

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

Offset: 0

Tanya Khovanova and Joshua Xiong, Apr 21 2014

nonn

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

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

Michael De Vlieger, Table of n, a(n) for n = 0..1024
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 37.
T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 7 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A241522 (4 piles), A241523 (5 piles).

Cf. A241717 (first differences).

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

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

A241718 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 the largest pile is n.

Original entry on oeis.org

1, 7, 13, 43, 25, 79, 133, 211, 49, 151, 253, 379, 457, 607, 757, 931, 97, 295, 493, 715, 889, 1135, 1381, 1651, 1681, 1975, 2269, 2587, 2857, 3199, 3541, 3907, 193, 583, 973, 1387, 1753, 2191, 2629, 3091, 3313, 3799, 4285, 4795, 5257, 5791, 6325

Offset: 0

Tanya Khovanova and Joshua Xiong, Apr 27 2014

nonn

This is the first difference of A241522.

			If the largest pile is 2, then there are 6 positions that are permutations of (0,0,2,2) plus 6 positions that are permutations of (1,1,2,2) and one position (2,2,2,2). Therefore, a(2)=13.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO], 2014, p. 8 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A241522, A241717 (3 piles), A241731 (5 piles).

Table[Length[Select[Flatten[Table[{n, k, j, BitXor[n, k, j]}, {n, 0, a}, {k, 0, a}, {j, 0, a}], 2], Max[#] == a &]], {a, 0, 50}]

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) = (12*c-6)*2^b + a(c-1).

A241731 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 the largest pile is n.

Original entry on oeis.org

1, 15, 45, 195, 165, 555, 1125, 1995, 645, 1995, 3525, 5355, 7605, 10395, 13845, 18075, 2565, 7755, 13125, 18795, 24885, 31515, 38805, 46875, 55845, 65835, 76965, 89355, 103125, 118395, 135285, 153915, 10245, 30795, 51525, 72555, 94005, 115995

Offset: 0

Tanya Khovanova and Joshua Xiong, Apr 27 2014

nonn

This is the finite difference of A241523.

			If the largest pile is 1, then there are 10 positions that are permutations of (0,0,0,1,1) plus 5 positions that are permutations of (0,1,1,1,1). Therefore, a(1)=15.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 9 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A241523, A241717 (3 piles), A241718 (4 piles).

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], Max[#] == a &]], {a, 0, 50}]

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)= 10*2^(2*b)*(2*c-1) + 20*c^3 - 30*c^2 + 20*c - 5.

A261695 First differences of A256534.

Original entry on oeis.org

0, 4, 12, 12, 36, 12, 36, 60, 84, 12, 36, 60, 84, 108, 132, 156, 180, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 12, 36, 60, 84, 108

Offset: 0

Omar E. Pol, Sep 24 2015

Number of cells turned ON at n-th stage of cellular automaton of A256534.

Similar to A256531 which shares infinitely many terms.

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
4;
12;
12, 36;
12, 36, 60, 84;
12, 36, 60, 84, 108, 132, 156, 180;
12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372;
...

Cf. A011782, A139251, A147582, A161411, A161415, A241717, A256531, A256534.

It appears that a(n) = 4 * A241717(n-1), n >= 1.

A256255 Triangle read by rows: T(n,k) = 6*k + 1, n>=0, 0<=k<=(2^n-1).

Original entry on oeis.org

1, 1, 7, 1, 7, 13, 19, 1, 7, 13, 19, 25, 31, 37, 43, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103

Offset: 0

Omar E. Pol, Apr 30 2015

Row n lists the first 2^n terms of A016921, n >= 0.
Row sums give A165665.
Right border gives A048488.
The sum of all terms of the first k rows gives A060867(k).
The product of the terms of the third row is equal to the Hardy-Ramanujan number: 1 * 7 * 13 * 19 = 1729.

			Triangle begins:
1;
1,7;
1,7,13,19;
1,7,13,19,25,31,37,43;
1,7,13,19,25,31,37,43,49,55,61,67,73,79,85,91;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n   a(n)             Compact diagram
------------------------------------------------------------------------
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
0    1      |_|_  |_ _ _  |_ _ _ _ _ _ _  |
1    1      | |_| |_ _  | |_ _ _ _ _ _  | |
2    7      |_ _ _|_  | | |_ _ _ _ _  | | |
3    1      | | | |_| | | |_ _ _ _  | | | |
4    7      | | |_ _ _| | |_ _ _  | | | | |
5   13      | |_ _ _ _ _| |_ _  | | | | | |
6   19      |_ _ _ _ _ _ _|_  | | | | | | |
7    1      | | | | | | | |_| | | | | | | |
8    7      | | | | | | |_ _ _| | | | | | |
9   13      | | | | | |_ _ _ _ _| | | | | |
10  19      | | | | |_ _ _ _ _ _ _| | | | |
11  25      | | | |_ _ _ _ _ _ _ _ _| | | |
12  31      | | |_ _ _ _ _ _ _ _ _ _ _| | |
13  37      | |_ _ _ _ _ _ _ _ _ _ _ _ _| |
14  43      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
For other diagrams of the same family see A241717 and A256258.

Paolo Xausa, Table of n, a(n) for n = 0..16382 (rows 0..13 of the triangle, flattened)

Cf. A000225, A001235, A016921, A048488, A060867, A165665, A241717, A256258.

With[{rows=7},Array[Range[1,6*2^#,6]&,rows,0]] (* Paolo Xausa, Sep 26 2023 *)

cp's OEIS Frontend

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

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A236305 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 each pile does not exceed n.

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A241718 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 the largest pile is n.

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A241731 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 the largest pile is n.

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A261695 First differences of A256534.

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A256255 Triangle read by rows: T(n,k) = 6*k + 1, n>=0, 0<=k<=(2^n-1).

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