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a(n) = 3^(2*n+1) - 3^(n+1) + 1: The left Aurifeuillian factor of 3^(6*n+3) + 1.
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%I #19 Aug 01 2015 10:40:44

%S 1,19,217,2107,19441,176419,1592137,14342347,129120481,1162202419,

%T 10460176057,94142647387,847287015121,7625592702019,68630363015977,

%U 617673353237227,5559060437415361,50031544711579219,450283904728735897,4052555149532191867

%N a(n) = 3^(2*n+1) - 3^(n+1) + 1: The left Aurifeuillian factor of 3^(6*n+3) + 1.

%C The corresponding right Aurifeuillian factor is A198410(n+2): 3^(6*n+3) + 1 = (3^(2*n+1) + 1) * a(n) * A198410(n+2).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cunningham_project">Cunningham Project</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (13, -39, 27).

%F a(n) = 13*a(n-1) - 39*a(n-2) + 27*a(n-3).

%F G.f.: (1 + 3*x)^2/((1 - x)*(1 - 3*x)*(1 - 9*x)).

%t Table[3^(2n+1) - 3^(n+1) + 1, {n, 0, 30}]

%t LinearRecurrence[{13,-39,27},{1,19,217},30] (* _Harvey P. Dale_, Mar 17 2013 *)

%o (PARI) Vec((1 + 3*x)^2/((1 - x)*(1 - 3*x)*(1 - 9*x)) + O(x^30)) \\ _Michel Marcus_, Feb 12 2015

%Y Cf. A092440, A085601, A198410, A220979-A220990.

%K nonn,easy

%O 0,2

%A _Stuart Clary_, Dec 27 2012