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%I #24 Sep 08 2022 08:45:10
%S -1,1,8,34,131,493,1844,6886,25703,95929,358016,1336138,4986539,
%T 18610021,69453548,259204174,967363151,3610248433,13473630584,
%U 50284273906,187663465043,700369586269,2613814880036,9754889933878,36405744855479,135868089488041
%N a(n) = 4*a(n-1) - a(n-2) + 3, with a(0) = -1, a(1) = 1.
%C Apart from the initial -1, these are the numbers k such that the triangular number k*(k + 1)/2 is the sum of three consecutive triangular numbers - see A129803. - _Brian Nowell_, Nov 03 2009
%H Vincenzo Librandi, <a href="/A082840/b082840.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).
%F a(n) = A001571(n) - 1. - _N. J. A. Sloane_, Nov 03 2009
%F G.f.: -(1 -6*x +2*x^2)/((1 - x)*(1 - 4*x + x^2)).
%F a(n) = -3/2 + (1/12)*( (a -2*b +5)*a^n + (b -2*a +5)*b^n ), with a = 2 + sqrt(3), b = 2 - sqrt(3):.
%F a(n) = -3/2 + (3/4)*A003500(n) - (1/4)*A003500(n-1).
%F a(n) = (1/2)*(A001834(n) - 3).
%F E.g.f.: ((1 + sqrt(3))*exp((2 + sqrt(3))*x) + (1 - sqrt(3))*exp((2 - sqrt(3))*x) - 6*exp(x))/4. - _Franck Maminirina Ramaharo_, Nov 12 2018
%t CoefficientList[Series[(-1+6x-2x^2)/((1-x)(1-4x+x^2)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Apr 15 2014 *)
%t LinearRecurrence[{5,-5,1}, {-1,1,8}, 30] (* _G. C. Greubel_, Feb 25 2019 *)
%o (PARI) is(n)=ispolygonal(3/2*n*(n+1)+4,3) || n==-1 \\ _Charles R Greathouse IV_, Apr 14 2014
%o (PARI) my(x='x+O('x^30)); Vec(-(1-6*x+2*x^2)/((1-x)*(1-4*x+x^2))) \\ _G. C. Greubel_, Feb 25 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( -(1-6*x+2*x^2)/((1-x)*(1-4*x+x^2)) )); // _G. C. Greubel_, Feb 25 2019
%o (Sage) (-(1-6*x+2*x^2)/((1-x)*(1-4*x+x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 25 2019
%o (GAP) a:=[-1,1,8];; for n in [4..30] do a[n]:=5*a[n-1]-5*a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Feb 25 2019
%Y Cf. A001571, A071954.
%K easy,sign
%O 0,3
%A Mario Catalani (mario.catalani(AT)unito.it), Apr 14 2003