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a(n) = (1 - (-5)^n)/6.
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%I #40 Feb 07 2024 12:30:02

%S 1,-4,21,-104,521,-2604,13021,-65104,325521,-1627604,8138021,

%T -40690104,203450521,-1017252604,5086263021,-25431315104,127156575521,

%U -635782877604,3178914388021,-15894571940104,79472859700521

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

%C q-integers for q = -5.

%C 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 Vincenzo Librandi, <a href="/A014986/b014986.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-4,5).

%F a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1).

%F G.f.: x/((1-x)*(1+5*x)). - _Bruno Berselli_, Dec 07 2011

%F a(n) = -4*a(n-1) + 5*a(n-2). - _Vincenzo Librandi_, Jun 19 2012

%F E.g.f.: (exp(x) - exp(-5*x))/6. - _G. C. Greubel_, May 26 2018

%p a:=n->sum ((-5)^j, j=0..n): seq(a(n), n=0..25); # _Zerinvary Lajos_, Dec 16 2008

%t LinearRecurrence[{-4,5},{1,-4},30] (* _Vincenzo Librandi_, Jun 19 2012 *)

%o (Sage) [gaussian_binomial(n,1,-5) for n in range(1,22)] # _Zerinvary Lajos_, May 28 2009

%o (PARI) a(n)=(1-(-5)^n)/6 \\ _Charles R Greathouse IV_, Dec 07 2011

%o (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 Cf. A077925, A014983, A014985, A014987, A014989, A014990, A014991, A014992, A014993, A014994.

%K sign,easy

%O 1,2

%A _Olivier GĂ©rard_

%E Better name from _Ralf Stephan_, Jul 14 2013