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%I #28 Feb 29 2024 15:51:30
%S 1,1,1,7,7,7,19,19,19,37,37,37,61,61,61,91,91,91,127,127,127,169,169,
%T 169,217,217,217,271,271,271,331,331,331,397,397,397,469,469,469,547,
%U 547,547,631,631,631,721,721,721,817,817,817,919,919,919,1027,1027,1027,1141,1141
%N Expansion of (1+4*x^3+x^6)/((1-x)*(1-x^3)^2).
%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 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F G.f.: (1+4*x^3+x^6)/((1-x)*(1-x^3)^2).
%F a(n) = a(n-1)+2*a(n-3)-2*a(n-4)-a(n-6)+a(n-7), n>7. - _Wesley Ivan Hurt_, Jun 23 2015
%F A003215 with each term repeated three times: a(n) = A003215(floor(n/3)). - _Robert Israel_, Jul 14 2015
%p f:= gfun:-rectoproc({q(n+3)-3*q(n+2)+3*q(n+1)-q(n), q(0) = 1, q(1) = 7, q(2) = 19},q(n),remember):
%p seq(f(i)$3, i=0..30); # _Robert Israel_, Jul 14 2015
%t CoefficientList[Series[(1 + 4*x^3 + x^6)/((1 - x)*(1 - x^3)^2), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Jun 23 2015 *)
%t LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 1, 1, 7, 7, 7, 19}, 60] (* _Vincenzo Librandi_, Jul 13 2015 *)
%t With[{c=LinearRecurrence[{3,-3,1},{1,7,19},20]},{c,c,c}]//Flatten//Sort (* _Harvey P. Dale_, Aug 03 2019 *)
%o (Magma) I:=[1,1,1,7,7,7,19]; [n le 7 select I[n] else Self(n-1)+2*Self(n-3)-2*Self(n-4)-Self(n-6)+Self(n-7): n in [1..70]]; // _Vincenzo Librandi_, Jul 13 2015
%Y Cf. A003215.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, Mar 29 2004
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