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A014986 a(n) = (1 - (-5)^n)/6.

%I A014986 #40 Feb 07 2024 12:30:02
%S A014986 1,-4,21,-104,521,-2604,13021,-65104,325521,-1627604,8138021,
%T A014986 -40690104,203450521,-1017252604,5086263021,-25431315104,127156575521,
%U A014986 -635782877604,3178914388021,-15894571940104,79472859700521
%N A014986 a(n) = (1 - (-5)^n)/6.
%C A014986 q-integers for q = -5.
%C A014986 Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - _Milan Janjic_, Jan 27 2010
%H A014986 Vincenzo Librandi, <a href="/A014986/b014986.txt">Table of n, a(n) for n = 1..1000</a>
%H A014986 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-4,5).
%F A014986 a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1).
%F A014986 G.f.: x/((1-x)*(1+5*x)). - _Bruno Berselli_, Dec 07 2011
%F A014986 a(n) = -4*a(n-1) + 5*a(n-2). - _Vincenzo Librandi_, Jun 19 2012
%F A014986 E.g.f.: (exp(x) - exp(-5*x))/6. - _G. C. Greubel_, May 26 2018
%p A014986 a:=n->sum ((-5)^j, j=0..n): seq(a(n), n=0..25); # _Zerinvary Lajos_, Dec 16 2008
%t A014986 LinearRecurrence[{-4,5},{1,-4},30] (* _Vincenzo Librandi_, Jun 19 2012 *)
%o A014986 (Sage) [gaussian_binomial(n,1,-5) for n in range(1,22)] # _Zerinvary Lajos_, May 28 2009
%o A014986 (PARI) a(n)=(1-(-5)^n)/6 \\ _Charles R Greathouse IV_, Dec 07 2011
%o A014986 (Magma) I:=[1, -4]; [n le 2 select I[n] else -4*Self(n-1)+5*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 19 2012
%Y A014986 Cf. A077925, A014983, A014985, A014987, A014989, A014990, A014991, A014992, A014993, A014994.
%K A014986 sign,easy
%O A014986 1,2
%A A014986 _Olivier Gérard_
%E A014986 Better name from _Ralf Stephan_, Jul 14 2013