A130102 E.g.f.: (e^x - x)^2.

1, 0, 2, 2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072, 8166, 16356, 32738, 65504, 131038, 262108, 524250, 1048536, 2097110, 4194260, 8388562, 16777168, 33554382, 67108812, 134217674, 268435400, 536870854, 1073741764, 2147483586

Offset: 0

Author

Paul Barry, May 07 2007

Keywords

• nonn
• easy

Comments

a(n) is the number of length n binary sequences in which no symbol occurs exactly once. (The Rosenthal formula takes 2^n for the total number of binary sequences and subtracts n for each sequence of length n with a single 0 or 1.) - Geoffrey Critzer, Dec 03 2011

From Ambrosio Valencia-Romero, Mar 08 2022: (Start)

a(n), for n > 1, is the number of pure Nash equilibria in the symmetric n-player two-strategy normal-form unanimity game. Let i be a player in set N = {1, 2, 3, ..., n} and s(i) in set S = {0, 1} be i's strategy. Then i's payoff, u(i), in this game is given by:

u(i) = 1 if s(1) = s(2) = ... = s(n-1) = s(n); otherwise, u(i) = 0.

Only two of the a(n) pure equilibria in this unanimity game are strict: s = <0, 0, ..., 0, 0> and s = <1, 1, ..., 1, 1>; these are the diagonal collective strategies where all actors obtain the payoff u(i) = 1.

The other a(n)-2 pure equilibria are weak and produce an individual payoff of u(i) = 0; these correspond to the collective strategy outcomes where more than one and fewer than n-1 individual strategies differ. For instance, for n = 4, the a(4)-2 = 6 weak pure equilibria are <0, 0, 1, 1>, <0, 1, 0, 1>, <0, 1, 1, 0>, <1, 0, 0, 1>, <1, 0, 1, 0>, and <1, 1, 0, 0>. (End)

Examples

			a(4) = 8 because there are 8 sequences on {0,1} such that neither 0 nor 1 occurs exactly once: {0,0,0,0}, {0,0,1,1}, {0,1,0,1}, {0,1,1,0}, {1,0,0,1}, {1,0,1,0}, {1,1,0,0}, {1,1,1,1}. - _Geoffrey Critzer_, Dec 03 2011
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A005803, A060576.

Programs

Magma

I:=[1, 0, 2, 2, 8, 22]; [n le 6 select I[n] else 4*Self(n-1)-5*Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 28 2012

Mathematica

a=Exp[x]-x; Range[0,20]! CoefficientList[Series[a^2, {x,0,20}], x] (* Geoffrey Critzer, Dec 03 2011 *)
CoefficientList[Series[1+2*x^2-2*x^3/((2*x-1)*(x-1)^2),{x,0,40}],x] (* Vincenzo Librandi, Jun 28 2012 *)

Formula

a(n) = 2^n - 2*n for n <> 2 (cf. A005803). - Rainer Rosenthal, Feb 14 2010.

E.g.f.: e^(2*x) - 2*x*e^x + x^2.

a(n) = Sum_{k=0..n} binomial(n,k)*A060576(k)*A060576(n-k).

G.f. 1 + 2*x^2 - 2*x^3/((2*x - 1)*(x - 1)^2). - R. J. Mathar, Dec 04 2011