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a(n) = 3*a(n-1) + 7*a(n-2).
19

%I #35 Sep 08 2022 08:44:40

%S 0,1,3,16,69,319,1440,6553,29739,135088,613437,2785927,12651840,

%T 57457009,260933907,1185000784,5381539701,24439624591,110989651680,

%U 504046327177,2289066543291,10395523920112,47210037563373

%N a(n) = 3*a(n-1) + 7*a(n-2).

%C Linear 2nd order recurrence.

%H Vincenzo Librandi, <a href="/A015524/b015524.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 (3,7).

%F From _R. J. Mathar_, Apr 21 2008: (Start)

%F O.g.f.: x/(1 - 3*x - 7*x^2).

%F a(n) = 14^n*(1/A^n -(-1)^n/B^n)/sqrt(37), where A = sqrt(37) - 3 = A010491 - 3 and B = sqrt(37) + 3 = A010491 + 3. (End)

%F a(n) = (7*(111+23*sqrt(37))*(1/2*(3+sqrt(37)))^n + (2553 + 431*sqrt(37)) * (1/2 (3-sqrt(37)))^n)/(518*(45+8*sqrt(37))). - _Harvey P. Dale_, Jul 04 2011

%t a[n_]:=(MatrixPower[{{1,3},{1,-4}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)

%t LinearRecurrence[{3,7},{0,1},30] (* _Harvey P. Dale_, Jul 04 2011 *)

%o (Sage) [lucas_number1(n,3,-7) for n in range(0, 23)] # _Zerinvary Lajos_, Apr 22 2009

%o (Magma) [n le 2 select n-1 else 3*Self(n-1)+7*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 12 2012

%o (PARI) x='x+O('x^30); concat([0], Vec(x/(1 - 3*x - 7*x^2))) \\ _G. C. Greubel_, Jan 01 2018

%K nonn,easy

%O 0,3

%A _Olivier GĂ©rard_