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 A015000 q-integers for q=-13. 4

%I

%S 1,-12,157,-2040,26521,-344772,4482037,-58266480,757464241,

%T -9847035132,128011456717,-1664148937320,21633936185161,

%U -281241170407092,3656135215292197,-47529757798798560,617886851384381281

%N q-integers for q=-13.

%H Vincenzo Librandi, <a href="/A015000/b015000.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-12,13).

%F a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1), with q=-13.

%F a(n) = Sum_{j=0..n-1} (-13)^j. - _Zerinvary Lajos_, Dec 16 2008

%F G.f.: x/((1 - x)*(1 + 13*x)). - _Vincenzo Librandi_, Oct 22 2012

%F a(n) = -12*a(n-1) + 13*a(n-2). - _Vincenzo Librandi_, Oct 22 2012

%F From _G. C. Greubel_, May 26 2018: (Start)

%F a(n) = (1 - (-13)^n)/14.

%F E.g.f.: (exp(x) - exp(-13*x))/14. (End)

%p a:=n->sum ((-13)^j, j=0..n-1): seq(a(n), n=0..20); # _Zerinvary Lajos_, Dec 16 2008

%t QBinomial[Range[20],1,-13] (* _Harvey P. Dale_, May 02 2012 *)

%t LinearRecurrence[{-12, 13}, {1, -12}, 30] (* _Vincenzo Librandi_, Oct 22 2012 *)

%o (Sage) [gaussian_binomial(n,1,-13) for n in range(1,18)] # _Zerinvary Lajos_, May 28 2009

%o (MAGMA) I:=[1,-12]; [n le 2 select I[n] else -12*Self(n-1)+13*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Oct 22 2012

%o (PARI) for(n=1, 30, print1((1-(-13)^n)/14, ", ")) \\ _G. C. Greubel_, May 26 2018

%Y Cf. A077925, A014983, A014985-A014987, A014989-A014994.

%K sign,easy

%O 1,2