A051253 - OEIS

%I #38 Jan 04 2025 17:28:55

%S 1,4,6,18,36,80,172,360,760,1576,3264,6720,13776,28160,57376,116640,

%T 236608,479104,968640,1955712,3944064,7945856,15993856,32168448,

%U 64656640,129879040,260759040,523289088,1049711616,2104967168,4219743232,8456841216,16944388096

%N Weights of rotation-symmetric functions in n variables.

%H A. Cabello, M. G. Parker, G. Scarpa and S. Severini, <a href="http://arxiv.org/abs/1211.4250">Exclusive disjunction structures and graph representatives of local complementation orbits</a>, arXiv preprint arXiv:1211.4250, 2012.

%H A. Cabello, M. G. Parker, G. Scarpa, and S. Severini, <a href="http://www.ii.uib.no/~matthew/AMGS0613gianni.pdf">Exclusivity structures and graph representatives of local complementation orbits</a>, (2013).

%H T. W. Cusick and P. Stanica, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00354-0">Fast Evaluation, Weights and Nonlinearity of Rotation-Symmetric Functions</a>, Discr. Math. 258 (2002), 289-301.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-4).

%F a(n+3) = 2*a(n+1)+2*a(n)+2^n.

%F G.f.: -x^3*(4*x^2-2*x-1)/((2*x^3+2*x^2-1)*(2*x-1)).

%F a(3)=1, a(4)=4, a(5)=6, a(6)=18, a(n) = 2*a(n-1)+2*a(n-2)-2*a(n-3)-4*a(n-4). - _Harvey P. Dale_, Mar 15 2015

%e a(3)=1 since the rotation-symmetric function x_1*x_2*x_3 has Hamming weight 1.

%e a(4)=4 since the rotation-symmetric function x_1*x_2*x_3+x_2*x_3*x_4+x_3*x_4*x_1+x_4*x_1*x_2 has Hamming weight 4.

%p t1:=(8*x^6/(1-2*x) + x^3 + 4*x^4 + 4*x^5)/(1-2*x^2-2*x^3);

%p t2:=series(t1,x,40);

%p seriestolist(%);

%t LinearRecurrence[{2,2,-2,-4},{1,4,6,18},40] (* _Harvey P. Dale_, May 05 2011 *)

%o (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -4,-2,2,2]^(n-3)*[1;4;6;18])[1,1] \\ _Charles R Greathouse IV_, Feb 19 2017

%K nice,easy,nonn,changed

%O 3,2

%A Pantelimon Stanica (stanpan(AT)sciences.aum.edu)

%E More terms from _Harvey P. Dale_, May 05 2011