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

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

Author

Ambrosio Valencia-Romero, Jan 14 2022

Keywords

• nonn
• easy
• tabl

Comments

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.

Examples

			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;
  ...
		

Links

• 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).

Crossrefs

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

Row sums give A145654.

Cf. A001047.

Programs

Maple

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

PARI

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

Formula

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