Joshua Xiong - cp's OEIS frontend

A241988 The number of P-positions in the Consecutive game with at most three piles, allowing for piles of zero, that are born by generation n.

1, 6, 15, 28, 41, 64, 93, 122, 151, 190, 229, 276, 337, 394, 445, 508, 581, 660, 741, 820, 903, 996, 1083, 1170, 1275, 1372, 1467, 1590, 1711, 1848, 1977, 2088, 2213, 2358, 2489, 2648, 2801, 2972, 3133, 3306

Offset: 0

Tanya Khovanova and Joshua Xiong, Aug 10 2014

nonn

In the Consecutive game, there are several piles of counters. A player is allowed to take the same positive number of counters from any subset of consecutive piles or any positive number of counters from one pile. The player who cannot move loses.

			For n = 1 the a(1) = 6 P-positions are (0,0,0), (1,0,1), (0,1,2), (0,2,1), (1,2,0), and (2,1,0).

T. Khovanova and J. Xiong, Cookie Monster Plays Games, arXiv:1407.1533 [math.HO], 2014.

Cf. A241987 (partial sums), A237686 (Nim), A241984 (Cookie Monster Game), A241986 (At-Most-2-Jars Game).

A241987 The number of P-positions in the Consecutive game with at most three piles, allowing for piles of zero, that are born in generation n.

Original entry on oeis.org

1, 5, 9, 13, 13, 23, 29, 29, 29, 39, 39, 47, 61, 57, 51, 63, 73, 79, 81, 79, 83, 93, 87, 87, 105, 97, 95, 123, 121, 137, 129, 111, 125, 145, 131, 159, 153, 171, 161, 173

Offset: 0

Tanya Khovanova and Joshua Xiong, Aug 10 2014

nonn

In the Consecutive game, there are several piles of counters. A player is allowed to take the same positive number of counters from any subset of consecutive piles or any positive number of counters from one pile. The player who cannot move loses.

a(n) is always odd.

			For n = 1 the a(1) = 5 P-positions are (1,0,1), (0,1,2), (0,2,1), (1,2,0), and (2,1,0).

T. Khovanova and J. Xiong, Cookie Monster Plays Games, arXiv:1407.1533 [math.HO], 2014.

Cf. A241988 (first differences), A237711 (Nim), A241983 (Cookie Monster Game), A241985 (At-Most-2-Jars Game).

A241986 The number of P-positions in the At-Most-2-Jars game with at most three piles, allowing for piles of zero, that are born by generation n.

Original entry on oeis.org

1, 8, 21, 34, 59, 96, 133, 176, 213, 256, 311, 360, 433, 512, 591, 700, 797, 912, 997, 1094, 1191, 1336, 1457, 1566, 1729, 1880, 2031, 2146, 2267, 2448, 2623, 2834, 3027, 3220, 3431, 3600, 3811, 4082, 4269, 4450

Offset: 0

Tanya Khovanova and Joshua Xiong, Aug 10 2014

nonn

In the At-Most-2-Jars game, there are several piles of counters. A player is allowed to take the same positive number of counters from any subset of two piles or any positive number of counters from one pile. The player who cannot move loses.

			For n = 1 the a(1) = 8 P-positions are (0,0,0), (1,1,1), and permutations of (0,1,2).

T. Khovanova and J. Xiong, Cookie Monster Plays Games, arXiv:1407.1533 [math.HO], 2014.

Cf. A241985 (partial sums), A237686 (Nim), A241984 (Cookie Monster Game), A241988 (Consecutive Game).

A241985 The number of P-positions in the At-Most-2-Jars game with up to three piles, allowing for piles of zero, that are born in generation n.

Original entry on oeis.org

1, 7, 13, 13, 25, 37, 37, 43, 37, 43, 55, 49, 73, 79, 79, 109, 97, 115, 85, 97, 97, 145, 121, 109, 163, 151, 151, 115, 121, 181, 175, 211, 193, 193, 211, 169, 211, 271, 187, 181

Offset: 0

Tanya Khovanova and Joshua Xiong, Aug 10 2014

nonn

In the At-Most-2-Jars game, there are several piles of counters. A player is allowed to take the same positive number of counters from any subset of two piles or any positive number of counters from one pile. The player who cannot move loses.

a(n) is always 1 greater than a multiple of 6.

			For n = 1 the a(1) = 7 P-positions are (1,1,1) and permutations of (0,1,2).

T. Khovanova and J. Xiong, Cookie Monster Plays Games, arXiv:1407.1533 [math.HO], 2014.

Cf. A241986 (first differences), A237711 (Nim), A241983 (Cookie Monster Game), A241987 (Consecutive Game).

A241984 The number of P-positions in the Cookie Monster game with at most three piles, allowing for piles of zero, that are born by generation n.

Original entry on oeis.org

1, 7, 19, 37, 55, 82, 127, 166, 232, 316, 385, 463, 547, 634, 706, 805, 922, 1036, 1165, 1294, 1429, 1597, 1735, 1888, 2041, 2203, 2395, 2596, 2749, 2911, 3133, 3337, 3559, 3772, 4009, 4261, 4489, 4723, 4987, 5242

Offset: 0

Tanya Khovanova and Joshua Xiong, Aug 10 2014

nonn

In the Cookie Monster game, there are several piles of counters. A player is allowed to take the same positive number of counters from any nonempty subset of the piles. The player who cannot move loses.

			For n = 1 the a(1) = 7 P-positions are (0,0,0) and (0,1,2), and permutations.

T. Khovanova and J. Xiong, Cookie Monster Plays Games, arXiv:1407.1533 [math.HO], 2014.

Cf. A241983 (partial sums), A237686 (Nim), A241986 (At-Most-2-Jars Game), A241988 (Consecutive Game).

A241983 The number of P-positions in generation n of the Cookie Monster game with up to three piles, allowing for piles of zero.

Original entry on oeis.org

1, 6, 12, 18, 18, 27, 45, 39, 66, 84, 69, 78, 84, 87, 72, 99, 117, 114, 129, 129, 135, 168, 138, 153, 153, 162, 192, 201, 153, 162, 222, 204, 222, 213, 237, 252, 228, 234, 264, 255

Offset: 0

Tanya Khovanova and Joshua Xiong, Aug 10 2014

nonn

			The a(1) = 6 P-positions are (0,1,2) and permutations.

A238759 The number of P-positions in the game of Nim with up to five piles, allowing for piles of zero, such that the total number of objects in all piles is 2n.

Original entry on oeis.org

1, 10, 15, 100, 65, 150, 175, 1000, 565, 650, 475, 1500, 925, 1750, 1875, 10000, 5565, 5650, 3475, 6500, 3725, 4750, 3875, 15000, 8425, 9250, 6375, 17500, 10625, 18750, 19375, 100000, 55565, 55650, 33475, 56500, 31725, 34750, 23875, 65000

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

First differences of A238147.

			The P-positions with the total of 4 are permutations of (0,0,0,2,2) and (0,1,1,1,1). Therefore, a(2)=15.

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

Cf. A238147 (partial sums), A048883 (3 piles), A237711 (4 piles), A241523, A241731.

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], Total[#] == a &]], {a, 0, 90, 2}]
(* Second program: *)
(* b = A238147 *) b[n_] := b[n] = Which[n <= 1, {1, 11}[[n+1]], OddQ[n], 11 b[(n-1)/2] + 5 b[(n-1)/2 - 1], EvenQ[n], b[(n-2)/2 + 1] + 15 b[(n-2)/2]];
Join[{1}, Differences[Array[b, 40, 0]]] (* Jean-François Alcover, Dec 14 2018 *)

a(2n+1) = 10*a(n), a(2n+2) = a(n+1) + 5*a(n).

A238147 The number of P-positions in the game of Nim with up to five piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

Original entry on oeis.org

1, 11, 26, 126, 191, 341, 516, 1516, 2081, 2731, 3206, 4706, 5631, 7381, 9256, 19256, 24821, 30471, 33946, 40446, 44171, 48921, 52796, 67796, 76221, 85471, 91846, 109346, 119971, 138721, 158096, 258096, 313661, 369311

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

Partial sums of A238759.

			There is 1 position (0,0,0,0,0) with a total of zero. There are 10 positions with a total of 2 that are permutations of (0,0,0,1,1). Therefore, a(1)=11.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 18 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A238759 (first differences), A130665 (3 piles), A237686 (4 piles), A241523, A241731.

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}]
(* Second program: *)
a[n_] := a[n] = Which[n <= 1, {1, 11}[[n+1]], OddQ[n], 11 a[(n-1)/2] + 5 a[(n-1)/2 - 1], EvenQ[n], a[(n-2)/2 + 1] + 15*a[(n-2)/2]];
Array[a, 34, 0] (* Jean-François Alcover, Dec 14 2018 *)

a(2n+1) = 11a(n) + 5a(n-1), a(2n+2) = a(n+1) + 15a(n).

A237711 The number of P-positions in the game of Nim with up to four piles, allowing for piles of zero, such that the total number of objects in all piles is 2n.

Original entry on oeis.org

1, 6, 7, 36, 13, 42, 43, 216, 49, 78, 55, 252, 85, 258, 259, 1296, 265, 294, 127, 468, 133, 330, 307, 1512, 337, 510, 343, 1548, 517, 1554, 1555, 7776, 1561, 1590, 559, 1764, 421, 762, 595, 2808, 601, 798, 463, 1980, 637, 1842, 1819, 9072, 1849

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

First differences of A237686.

			The P-positions with the total of 4 are permutations of (0,0,2,2) and (1,1,1,1). Therefore, a(2)=7.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 16 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A237686 (partial sums), A048883 (3 piles), A238759 (5 piles), A241522, A241718.

Table[Length[
  Select[Flatten[
    Table[{n, k, j, BitXor[n, k, j]}, {n, 0, a}, {k, 0, a}, {j, 0,
      a}], 2], Total[#] == a &]], {a, 0, 100, 2}]

a(2n+1) = 6a(n), a(2n+2) = a(n+1) + a(n).

G.f.: Product_{k>=0} (1 + 6*x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Mar 16 2021

A237686 The number of P-positions in the game of Nim with up to four piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

Original entry on oeis.org

1, 7, 14, 50, 63, 105, 148, 364, 413, 491, 546, 798, 883, 1141, 1400, 2696, 2961, 3255, 3382, 3850, 3983, 4313, 4620, 6132, 6469, 6979, 7322, 8870, 9387, 10941, 12496, 20272, 21833, 23423, 23982, 25746, 26167, 26929, 27524, 30332, 30933

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

Partial sums of A237711.

			There is a position (0,0,0,0) with a total of zero. There are 6 positions with a total of 2 that are permutations of (0,0,1,1). Therefore, a(1)=7.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 16 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A237711 (first differences), A130665 (3 piles), A238147 (5 piles), A241522, A241718.

Table[Length[
  Select[Flatten[
    Table[{n, k, j, BitXor[n, k, j]}, {n, 0, a}, {k, 0, a}, {j, 0,
      a}], 2], Total[#] <= a &]], {a, 0, 100, 2}]

a(2n+1) = 7a(n) + a(n-1), a(2n+2) = a(n+1) + 7a(n).

cp's OEIS Frontend

User: Joshua Xiong

A241988 The number of P-positions in the Consecutive game with at most three piles, allowing for piles of zero, that are born by generation n.

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A241987 The number of P-positions in the Consecutive game with at most three piles, allowing for piles of zero, that are born in generation n.

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A241986 The number of P-positions in the At-Most-2-Jars game with at most three piles, allowing for piles of zero, that are born by generation n.

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A241985 The number of P-positions in the At-Most-2-Jars game with up to three piles, allowing for piles of zero, that are born in generation n.

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A241984 The number of P-positions in the Cookie Monster game with at most three piles, allowing for piles of zero, that are born by generation n.

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A241983 The number of P-positions in generation n of the Cookie Monster game with up to three piles, allowing for piles of zero.

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A238759 The number of P-positions in the game of Nim with up to five piles, allowing for piles of zero, such that the total number of objects in all piles is 2n.

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A238147 The number of P-positions in the game of Nim with up to five piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

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A237711 The number of P-positions in the game of Nim with up to four piles, allowing for piles of zero, such that the total number of objects in all piles is 2n.

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