Ambrosio Valencia-Romero - cp's OEIS frontend

User: Ambrosio Valencia-Romero

Ambrosio Valencia-Romero has authored 5 sequences.

A357298 Triangle read by rows where all entries in every even row are 1's and the entries in every odd row alternate between 0 (start/end) and 1.

0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Ambrosio Valencia-Romero, Dec 20 2022

Row sums are equal to n for even n and (n-1)/2 for odd n; or A065423(n+1).

			Triangle begins:
   n\k  0  1  2  3  4  5  6  7  8  9 ...
   1    0;
   2    1, 1;
   3    0, 1, 0;
   4    1, 1, 1, 1;
   5    0, 1, 0, 1, 0;
   6    1, 1, 1, 1, 1, 1;
   7    0, 1, 0, 1, 0, 1, 0;
   8    1, 1, 1, 1, 1, 1, 1, 1;
   9    0, 1, 0, 1, 0, 1, 0, 1, 0;
  10    1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ...
Formatted as a symmetric triangle -- regular hexagram pattern with 0's at the centers formed by connecting all 1's:
      .----------------------------------------------.
      |                      k=0   1   2   3   4   5 |
      |-----------------------/---/---/---/---/--./  |
-------                          /   /   /   /   /   |
| n=1 |                     0   /   /   /   /   /|   |
-------                            /   /   /   / | 6 |
|   2 |                   1---1   /   /   /   /  |/  |
-------                    \ /       /   /   /   /   |
|   3 |                 0   1   0   /   /   /   /|   |
-------                    / \         /   /   / | 7 |
|   4 |               1---1---1---1   /   /   /  |/  |
-------                \ /     \ /       /   /   /   |
|   5 |             0   1   0   1   0   /   /   /|   |
-------                / \     / \         /   / | 8 |
|   6 |           1---1---1---1---1---1   /   /  |/  |
-------            \ /     \ /     \ /       /   /   |
|   7 |         0   1   0   1   0   1   0   /   /|   |
-------            / \     / \     / \         / | 9 |
|   8 |       1---1---1---1---1---1---1---1   /   /  |
-------        \ /     \ /     \ /     \ /       /   |
|   9 |     0   1   0   1   0   1   0   1   0   /|   |
-------        / \     / \     / \     / \       | . |
|  10 |   1---1---1---1---1---1---1---1---1---1  | . |
-------                                          | . |

Cf. A358125, A065423 (row sums).

T := n -> local k; seq(1/2 + 1/2*(-1)^(n*(k + 1)), k = 0 .. n - 1); # formula 1
seq(T(n), n=1..16); # print first 16 rows of formula 1.

T(n,k) = bitnegimply(1,n) || bitand(1,k); \\ Kevin Ryde, Dec 21 2022

T(n, k) = 1/2 + (1/2)*(-1)^(n*(k+1)), for n >= 1 and 0 <= k <= n-1.
T(n, k) = (2^n - 2^(n-k-1) - 2^k) mod 3, for n >= 1 and 0 <= k <= n-1.
T(n, k) = A358125(n, k) mod 3, for n >= 1 and 0 <= k <= n-1.

A359200 Triangle read by rows: T(n, k) = A358125(n,k)*binomial(n-1, k), 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 8, 3, 7, 30, 30, 7, 15, 88, 144, 88, 15, 31, 230, 520, 520, 230, 31, 63, 564, 1620, 2240, 1620, 564, 63, 127, 1330, 4620, 8120, 8120, 4620, 1330, 127, 255, 3056, 12432, 26432, 33600, 26432, 12432, 3056, 255, 511, 6894, 32112, 79968, 122976, 122976, 79968, 32112, 6894, 511

Offset: 1

Ambrosio Valencia-Romero, Dec 20 2022

			Triangle begins:
   0;
   1,     1;
   3,     8,     3;
   7,    30,    30,      7;
  15,    88,   144,     88,     15;
  31,   230,   520,    520,    230,     31;
  63,   564,  1620,   2240,   1620,    564,     63;
 127,  1330,  4620,   8120,   8120,   4620,   1330,    127;
 255,  3056, 12432,  26432,  33600,  26432,  12432,   3056,   255;
 511,  6894, 32112,  79968, 122976, 122976,  79968,  32112,  6894,   511;
1023, 15340, 80460, 229440, 413280, 499968, 413280, 229440, 80460, 15340, 1023;
...

Row sums give 2*A005061(n-1).

T := n -> local k; seq((2^n - 2^(n - k - 1) - 2^k)*binomial(n - 1, k), k = 0 .. n - 1);
seq(T(n), n = 1 .. 11);

T[n_, k_] := (2^n - 2^(n - k - 1) - 2^k)*Binomial[n - 1, k]; Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, Dec 20 2022 *)

T(n, k) = (2^n - 2^(n-k-1) - 2^k)*binomial(n-1, k), for n >= 1 and 0 <= k <= n-1.

A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 7, 10, 10, 7, 15, 22, 24, 22, 15, 31, 46, 52, 52, 46, 31, 63, 94, 108, 112, 108, 94, 63, 127, 190, 220, 232, 232, 220, 190, 127, 255, 382, 444, 472, 480, 472, 444, 382, 255, 511, 766, 892, 952, 976, 976, 952, 892, 766, 511, 1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023

Offset: 1

Ambrosio Valencia-Romero, Dec 20 2022

T(n, k) is the expanded number of player-reduced static games within an n-player two-strategy game scenario in which one player (the "standpoint") faces a specific combination of other players' individual strategies with the possibility of anti-coordination between them -- the total number of such combinations is 2^(n-1). The value of k represents the number of other players who (are expected to) agree on one of the two strategies in S, while the other n-k-1 choose the other strategy; the standpoint player is not included.

			Triangle begins:
  0;
  1,     1;
  3,     4,    3;
  7,    10,   10,    7;
  15,   22,   24,   22,   15;
  31,   46,   52,   52,   46,   31;
  63,   94,  108,  112,  108,   94,   63;
 127,  190,  220,  232,  232,  220,  190,  127;
 255,  382,  444,  472,  480,  472,  444,  382,  255;
 511,  766,  892,  952,  976,  976,  952,  892,  766,  511;
1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023;
2047, 3070, 3580, 3832, 3952, 4000, 4000, 3952, 3832, 3580, 3070, 2047;
  ...

Column k=0 gives A000225(n-1).
Column k=1 gives A033484(n-2).
Column k=2 gives A053208(n-3).
Cf. A005061, A359200.

T := n -> seq(2^n - 2^(n - k - 1) - 2^k, k = 0 .. n - 1);
seq(T(n), n=1..12);

T[n_, k_] := 2^n - 2^(n - k - 1) - 2^k; Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, Dec 20 2022 *)

T(n, k) = 2^n - 2^(n-k-1) - 2^k.

Sum_{k=0..n-1} T(n,k)*binomial(n-1,k) = 2*A005061(n-1)

A350771 Triangle read by rows: T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k), 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 7, 12, 12, 7, 15, 32, 36, 32, 15, 31, 80, 100, 100, 80, 31, 63, 192, 270, 280, 270, 192, 63, 127, 448, 714, 770, 770, 714, 448, 127, 255, 1024, 1848, 2128, 2100, 2128, 1848, 1024, 255, 511, 2304, 4680, 5880, 5796, 5796, 5880, 4680, 2304, 511, 1023, 5120, 11610, 16080, 16380, 15624, 16380, 16080, 11610, 5120, 1023

Offset: 1

Ambrosio Valencia-Romero, Jan 14 2022

The elements in T(n,k) result from the product of each element of A350770(n,k) and binomial(n-1,k).

			Triangle begins:
     0;
     1,    1;
     3,    4,     3;
     7,   12,    12,     7;
    15,   32,    36,    32,    15;
    31,   80,   100,   100,    80,    31;
    63,  192,   270,   280,   270,   192,    63;
   127,  448,   714,   770,   770,   714,   448,   127;
   255, 1024,  1848,  2128,  2100,  2128,  1848,  1024,   255;
   511, 2304,  4680,  5880,  5796,  5796,  5880,  4680,  2304,  511;
  1023, 5120, 11610, 16080, 16380, 15624, 16380, 16080, 11610, 5120, 1023;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Ambrosio Valencia-Romero and P. T. Grogan, The strategy dynamics of collective systems: Underlying hindrances beyond two-actor coordination, PLOS ONE 19(4): e0301394 (S1 Appendix).

Column k=0 gives A000225(n-1).

Row sums give A056182(n-1) = 2*A001047(n-1).

T := n -> local k; seq((2^(n - k - 1) + 2^k - 2)*binomial(n - 1, k), k = 0 .. n - 1);
seq(T(n), n = 1 .. 11);

T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k) \\ Andrew Howroyd, Jan 05 2024

T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k).

A350770 Triangle read by rows: T(n, k) = 2^(n-k-1) + 2^k - 2, 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 7, 4, 4, 7, 15, 8, 6, 8, 15, 31, 16, 10, 10, 16, 31, 63, 32, 18, 14, 18, 32, 63, 127, 64, 34, 22, 22, 34, 64, 127, 255, 128, 66, 38, 30, 38, 66, 128, 255, 511, 256, 130, 70, 46, 46, 70, 130, 256, 511, 1023, 512, 258, 134, 78, 62, 78, 134, 258, 512, 1023, 2047, 1024, 514, 262, 142, 94, 94, 142, 262, 514, 1024, 2047

Offset: 1

Ambrosio Valencia-Romero, Jan 14 2022

T(n, k) is the number of player-reduced static games within an n-player two-strategy game scenario in which one player (the "standpoint") faces a specific combination of other players' individual strategies without the possibility of anti-coordination between them -- the total number of such combinations is 2^(n-1). The value of k represents the number of other players who (are expected to) agree on one of the two strategies in S, while the other n-k-1 choose the other strategy; the standpoint player is not included.

The sum of the products of T(n, k) and binomial(n-1,k) for 0 <= k <= n-1 equals 2*A001047(n-1). For instance, for n = 3, T(3, k) returns 3, 2, and 3 and binomial(3-1,k) returns 1, 2, and 1 for k = 0, 1, and 2, respectively. Then 3*1 + 2*2 + 3*1 = 2*A001047(3-1) = 2*5 = 10. Similarly, for n = 4, the result yields 7*1 + 4*3 + 4*3 + 7*1 = 2*A001047(4-1) = 2*19 = 38.

			Triangle begins:
     0;
     1,    1;
     3,    2,   3;
     7,    4,   4,   7;
    15,    8,   6,   8,  15;
    31,   16,  10,  10,  16, 31;
    63,   32,  18,  14,  18, 32, 63;
   127,   64,  34,  22,  22, 34, 64, 127;
   255,  128,  66,  38,  30, 38, 66, 128, 255;
   511,  256, 130,  70,  46, 46, 70, 130, 256, 511;
  1023,  512, 258, 134,  78, 62, 78, 134, 258, 512, 1023;
  2047, 1024, 514, 262, 142, 94, 94, 142, 262, 514, 1024, 2047;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50).
Ambrosio Valencia-Romero, Strategy Dynamics in Collective Systems Design, Ph.D. Thesis, Stevens Institute of Technology (Hoboken, 2021). [Table 5.4, page 67]
Ambrosio Valencia-Romero and P. T. Grogan, The strategy dynamics of collective systems: Underlying hindrances beyond two-actor coordination, PLOS ONE 19(4): e0301394 (S1 Appendix).

Column k=0 gives A000225(n-1).
Row sums give A145654.
Cf. A001047.

T := n -> seq(2^(n - k - 1) + 2^k - 2, k = 0 .. n - 1);
seq(T(n), n=1..12);

T(n, k) = 2^(n-k-1) + 2^k - 2 \\ Andrew Howroyd, May 06 2023

T(n, k) = 2^(n-k-1) + 2^k - 2.

cp's OEIS Frontend

User: Ambrosio Valencia-Romero

A357298 Triangle read by rows where all entries in every even row are 1's and the entries in every odd row alternate between 0 (start/end) and 1.

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A359200 Triangle read by rows: T(n, k) = A358125(n,k)*binomial(n-1, k), 0 <= k <= n-1.

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A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.

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A350771 Triangle read by rows: T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k), 0 <= k <= n-1.

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A350770 Triangle read by rows: T(n, k) = 2^(n-k-1) + 2^k - 2, 0 <= k <= n-1.

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Examples

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Programs

Maple

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