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%I #30 Apr 17 2022 10:08:09
%S 1,0,2,2,8,22,52,114,240,494,1004,2026,4072,8166,16356,32738,65504,
%T 131038,262108,524250,1048536,2097110,4194260,8388562,16777168,
%U 33554382,67108812,134217674,268435400,536870854,1073741764,2147483586
%N E.g.f.: (e^x - x)^2.
%C 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
%C From _Ambrosio Valencia-Romero_, Mar 08 2022: (Start)
%C 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:
%C u(i) = 1 if s(1) = s(2) = ... = s(n-1) = s(n); otherwise, u(i) = 0.
%C 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.
%C 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)
%H Vincenzo Librandi, <a href="/A130102/b130102.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F a(n) = 2^n - 2*n for n <> 2 (cf. A005803). - _Rainer Rosenthal_, Feb 14 2010.
%F E.g.f.: e^(2*x) - 2*x*e^x + x^2.
%F a(n) = Sum_{k=0..n} binomial(n,k)*A060576(k)*A060576(n-k).
%F G.f. 1 + 2*x^2 - 2*x^3/((2*x - 1)*(x - 1)^2). - _R. J. Mathar_, Dec 04 2011
%e 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
%t a=Exp[x]-x; Range[0,20]! CoefficientList[Series[a^2, {x,0,20}], x] (* _Geoffrey Critzer_, Dec 03 2011 *)
%t CoefficientList[Series[1+2*x^2-2*x^3/((2*x-1)*(x-1)^2),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 28 2012 *)
%o (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
%Y Cf. A005803, A060576.
%K nonn,easy
%O 0,3
%A _Paul Barry_, May 07 2007
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