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%I #24 Sep 08 2022 08:45:23
%S 2,6,10,54,202,822,3274,13110,52426,209718,838858,3355446,13421770,
%T 53687094,214748362,858993462,3435973834,13743895350,54975581386,
%U 219902325558,879609302218,3518437208886,14073748835530,56294995342134,225179981368522,900719925474102
%N a(n) = 3*a(n-1) + 4*a(n-2), with a(0) = 2, a(1) = 6, a(2) = 10.
%H G. C. Greubel, <a href="/A115113/b115113.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,4).
%F From _Colin Barker_, Nov 13 2012: (Start)
%F a(n) = (-2*(7*(-1)^n - 2^(1 + 2*n)))/5 for n > 0.
%F a(n) = 3*a(n-1) + 4*a(n-2) for n > 2.
%F G.f.: 2*(8*x^2 - 1)/((x + 1)*(4*x - 1)). (End)
%F E.g.f.: (20 - 14*exp(-x) + 4*exp(4*x))/5. - _Franck Maminirina Ramaharo_, Nov 23 2018
%t Join[{2}, LinearRecurrence[{3, 4}, {6, 10}, 50]]
%o (Maxima) (a[0] : 2, a[1] : 6, a[2] : 10, a[n] := 3*a[n-1] + 4*a[n-2], makelist(a[n], n, 0, 50)); /* _Franck Maminirina Ramaharo_, Nov 23 2018 */
%o (PARI) x='x+O('x^50); Vec(2*(8*x^2-1)/((x+1)*(4*x-1))) \\ _G. C. Greubel_, Nov 23 2018
%o (Magma) I:=[6,10]; [2] cat [n le 2 select I[n] else 3*Self(n-1) + 4*Self(n-2): n in [1..49]]; // _G. C. Greubel_, Nov 23 2018
%o (Sage) s=(2*(8*x^2-1)/((x+1)*(4*x-1))).series(x,50); s.coefficients(x, sparse=False) # _G. C. Greubel_, Nov 23 2018
%Y Cf. A115164, A115335.
%K nonn,easy
%O 0,1
%A _Roger L. Bagula_, Mar 06 2006
%E Edited, and new name from _Franck Maminirina Ramaharo_, Nov 23 2018, after _Colin Barker_'s formula
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