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A014990 a(n) = (1 - (-8)^n)/9.

%I A014990 #40 Feb 10 2024 11:34:54
%S A014990 1,-7,57,-455,3641,-29127,233017,-1864135,14913081,-119304647,
%T A014990 954437177,-7635497415,61083979321,-488671834567,3909374676537,
%U A014990 -31274997412295,250199979298361,-2001599834386887,16012798675095097
%N A014990 a(n) = (1 - (-8)^n)/9.
%C A014990 q-integers for q=-8.
%H A014990 Vincenzo Librandi, <a href="/A014990/b014990.txt">Table of n, a(n) for n = 1..1000</a>
%H A014990 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-7,8).
%F A014990 a(n) = a(n-1) + q^{(n-1)} = {(q^n - 1) / (q - 1)}
%F A014990 From _Philippe Deléham_, Feb 13 2007: (Start)
%F A014990 a(1)=1, a(2)=-7, a(n) = -7*a(n-1) + 8*a(n-2) for n > 2.
%F A014990 a(n) = (-1)^(n+1)*A015565(n).
%F A014990 G.f.: x/(1 + 7*x - 8*x^2). (End)
%F A014990 E.g.f.: (exp(x) - exp(-8*x))/9. - _G. C. Greubel_, May 26 2018
%p A014990 a:=n->sum ((-8)^j, j=0..n): seq(a(n), n=0..25); # _Zerinvary Lajos_, Dec 16 2008
%t A014990 QBinomial[Range[20],1,-8] (* or *) LinearRecurrence[{-7,8},{1,-7},20] (* _Harvey P. Dale_, Dec 19 2011 *)
%o A014990 (Sage) [gaussian_binomial(n,1,-8) for n in range(1,20)] # _Zerinvary Lajos_, May 28 2009
%o A014990 (Magma) I:=[1, -7]; [n le 2 select I[n] else -7*Self(n-1) +8*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Oct 22 2012
%o A014990 (PARI) a(n)=(1-(-8)^n)/9 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A014990 Cf. A015565, A077925, A014983, A014985, A014986, A014987, A014989, A014991, A014992, A014993, A014994.
%K A014990 sign,easy
%O A014990 1,2
%A A014990 _Olivier Gérard_
%E A014990 Better name from _Ralf Stephan_, Jul 14 2013