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%I #40 Mar 15 2024 21:28:04
%S 1,-2,4,-10,27,-72,189,-494,1292,-3382,8855,-23184,60697,-158906,
%T 416020,-1089154,2851443,-7465176,19544085,-51167078,133957148,
%U -350704366,918155951,-2403763488,6293134513,-16475640050,43133785636,-112925716858,295643364939,-774004377960
%N Expansion of (1 + x)^2/((1 + x + x^2)*(1 + 3*x + x^2)).
%C Binomial transform gives signed version of A093040.
%C The positive sequence has g.f. (1 - x)^2/((1 - x + x^2)(1 - 3*x + x^2)) and a(n) = Sum_{k=0..n} binomial(n+k+1, n-k)*(1+(-1)^k)/2. - _Paul Barry_, Jul 06 2009
%C Floretion Algebra Multiplication Program, FAMP Code: 2basei[C*F]; C = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki'; F = + .5'i + .5'ii' + .5'ij' + .5'ik'
%D C. Dement, Floretion Integer Sequences (work in progress).
%H Muniru A Asiru, <a href="/A113066/b113066.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-5,-4,-1).
%F a(n) + a(n+1) = (-1)^(n+1)*A109961(n+1).
%F a(n) + a(n+1) + a(n+2) = (-1)^n*A001906(n+2) = (-1)^n*F(2*n+4).
%F a(n) = A049347(n)/2 + (-1)^n*A001906(n+1)/2. - _R. J. Mathar_, Nov 10 2009
%F Lim_{n -> inf} a(n)/a(n-1) = -(1 + A001622). - _A.H.M. Smeets_, Sep 11 2018
%F a(n) = -4*a(n-1) - 5*a(n-2) - 4*a(n-3) - a(n-4). - _Muniru A Asiru_, Sep 11 2018
%p seq(coeff(series((1+x)^2/((1+x+x^2)*(1+3*x+x^2)),x,n+1), x, n), n = 0 .. 35); # _Muniru A Asiru_, Sep 11 2018
%t LinearRecurrence[{-4, -5, -4, -1}, {1, -2, 4, -10}, 40] (* _Vincenzo Librandi_, Sep 12 2018 *)
%t CoefficientList[Series[(1 + x)^2/((1 + x + x^2)*(1 + 3 x + x^2)), {x, 0, 50}], x] (* _Stefano Spezia_, Sep 12 2018 *)
%o (PARI) x='x+O('x^99); Vec((1+x)^2/((1+x+x^2)*(1+3*x+x^2))) \\ _Altug Alkan_, Sep 11 2018
%o (GAP) a:=[1,-2,4,-10];; for n in [5..35] do a[n]:=-4*a[n-1]-5*a[n-2]-4*a[n-3]-a[n-4]; od; a; # _Muniru A Asiru_, Sep 11 2018
%o (Magma) I:=[1,-2,4,-10]; [n le 4 select I[n] else -4*Self(n-1)-5*Self(n-2)- 4*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Sep 12 2018
%Y Cf. A113067, A113068, A093040, A109961, A001906.
%K sign,easy
%O 0,2
%A _Creighton Dement_, Oct 13 2005
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