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%I #28 Jan 05 2024 07:26:44
%S 1,1,2,3,5,6,9,11,15,18,23,27,33,38,45,51,59,66,75,83,93,102,113,123,
%T 135,146,159,171,185,198,213,227,243,258,275,291,309,326,345,363,383,
%U 402,423,443,465,486,509,531,555
%N Molien series for ring of symmetrized weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).
%H G. C. Greubel, <a href="/A028309/b028309.txt">Table of n, a(n) for n = 0..1000</a>
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (<a href="http://neilsloane.com/doc/self.txt">Abstract</a>, <a href="http://neilsloane.com/doc/self.pdf">pdf</a>, <a href="http://neilsloane.com/doc/self.ps">ps</a>).
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F G.f.: (1 - x + x^3 - x^5 + x^6)/((1-x)^2 * (1-x^2)). - _Ralf Stephan_, Apr 29 2014
%F a(n) = (1/8) * (2*n^2 + 3*(-1)^n - 4*n + 21) for n >= 3. - _Ralf Stephan_, Apr 29 2014 [Corrected by _Pontus von Brömssen_, May 30 2023]
%F From _G. C. Greubel_, Jan 05 2024: (Start)
%F a(n) = (1/8)*(2*n^2 - 4*n + 21 + 3*(-1)^n) - 2*[n=0] - [n=1] - [n=2].
%F E.g.f.: (1/8)*( (21 - 2*x + 2*x^2)*exp(x) + 3*exp(-x) ) - (2 + x + x^2/2). (End)
%t LinearRecurrence[{2,0,-2,1},{1,1,2,3,5,6,9},50] (* _Harvey P. Dale_, Nov 06 2016 *)
%o (Magma) [n le 2 select Floor((n+2)/2) else (2*n^2-4*n+21+3*(-1)^n)/8: n in [0..50]]; // _G. C. Greubel_, Jan 05 2024
%o (SageMath) [(2*n^2-4*n+21+3*(-1)^n)/8 - ((4-n)//2)*int(n<3) for n in range(51)] # _G. C. Greubel_, Jan 05 2024
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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