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a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), starting with 1, 2, 6, 20.
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%I #19 Jan 01 2024 23:50:47

%S 1,2,6,20,61,183,547,1640,4920,14762,44287,132861,398581,1195742,

%T 3587226,10761680,32285041,96855123,290565367,871696100,2615088300,

%U 7845264902,23535794707,70607384121,211822152361,635466457082

%N a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), starting with 1, 2, 6, 20.

%C A132868(n) - a(n) = A128834(n) (discovered in 1995).

%H G. C. Greubel, <a href="/A132353/b132353.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-1,3).

%F Also a(n) - 3^(n+1) = hexaperiodic 1, -1, -3, -1, 1, 3; cf. A132951.

%F From _R. J. Mathar_, Apr 04 2008: (Start)

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

%F a(n) = -(-1)^n/12 + 3^(n+1)/4 + A057079(n+2)/3. (End)

%t LinearRecurrence[{3, 0, -1, 3}, {1, 2, 6, 20}, 50] (* _G. C. Greubel_, Jan 15 2018 *)

%o (PARI) x='x+O('x^30); Vec((1-x+3*x^3)/((1-3*x)*(1+x)*(x^2-x+1))) \\ _G. C. Greubel_, Jan 15 2018

%o (Magma) I:=[1,2,6,20]; [n le 4 select I[n] else 3*Self(n-1) - Self(n-3) + 3*Self(n-4): n in [1..30]]; // _G. C. Greubel_, Jan 15 2018

%Y Cf. A129339.

%K nonn

%O 0,2

%A _Paul Curtz_, Nov 24 2007

%E More terms from _R. J. Mathar_, Apr 04 2008