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A192873 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments.

%I A192873 #33 Sep 08 2022 08:45:58
%S A192873 0,1,2,7,18,49,128,337,882,2311,6050,15841,41472,108577,284258,744199,
%T A192873 1948338,5100817,13354112,34961521,91530450,239629831,627359042,
%U A192873 1642447297,4299982848,11257501249,29472520898,77160061447,202007663442,528862928881,1384581123200
%N A192873 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments.
%C A192873 The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = x*p(n-1,x) + (x^2)*p(n-1,x) + 1.  See A192872.
%C A192873 First differences give A236428. - _Richard R. Forberg_, Feb 23 2014
%H A192873 G. C. Greubel, <a href="/A192873/b192873.txt">Table of n, a(n) for n = 0..1000</a>
%H A192873 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).
%F A192873 a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
%F A192873 G.f.: x*(x^2-x+1) / ((1-x)*(1+x)*(x^2-3*x+1)). - _Colin Barker_, Apr 01 2014
%F A192873 a(n) = (1/10) * (4L(2*n) - 3*(-1)^n - 5), with L(n) the Lucas numbers (A000032). - _Ralf Stephan_, Apr 06 2014
%F A192873 a(-n) = a(n) for all n in Z. - _Michael Somos_, Apr 08 2014
%e A192873 The coefficients of all the polynomials p(n,x) are Fibonacci numbers (A000045).  The first 6 and their reductions:
%e A192873 p(0,x) = 1 -> 1
%e A192873 p(1,x) = x -> x
%e A192873 p(2,x) = 1 +2*x^2 -> 3 +2*x
%e A192873 p(3,x) = 1 +x +3*x^3 -> 4 +7*x
%e A192873 p(4,x) = 1 +x +2*x^2 +5*x^4 -> 13 +18*x
%e A192873 p(5,x) = 1 +x +2*x^2 +3*x^3 +8*x^5 -> 30 +49*x
%e A192873 G.f. = x + 2*x^2 + 7*x^3 + 18*x^4 + 49*x^5 + 128*x^6 + 337*x^7 + ...
%p A192873 seq(coeff(series(x*(x^2-x+1)/((1-x)*(1+x)*(x^2-3*x+1)),x,n+1), x, n), n = 0 .. 35); # _Muniru A Asiru_, Jan 08 2019
%t A192873 (See A192872.)
%t A192873 a[ n_] := SeriesCoefficient[ x * (1 - x + x^2) / ((1 - x^2) * (1 - 3*x + x^2)), {x, 0, Abs @ n}]; (* _Michael Somos_, Apr 08 2014 *)
%t A192873 LinearRecurrence[{3,0,-3,1}, {0,1,2,7}, 40] (* _G. C. Greubel_, Jan 07 2019 *)
%o A192873 (PARI) concat(0, Vec(-x*(x^2-x+1)/((x-1)*(x+1)*(x^2-3*x+1)) + O(x^40))) \\ _Colin Barker_, Apr 01 2014
%o A192873 (Magma) I:=[0,1,2,7]; [n le 4 select I[n] else 3*Self(n-1) - 3*Self(n-3) +Self(n-4): n in [1..40]]; // _G. C. Greubel_, Jan 07 2019
%o A192873 (Sage) (x*(x^2-x+1)/((1-x^2)*(x^2-3*x+1))).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 07 2019
%o A192873 (GAP) a:=[0,1,2,7];; for n in [5..40] do a[n]:=3*a[n-1]-3*a[n-3]+a[n-4]; od; a; # _G. C. Greubel_, Jan 07 2019
%Y A192873 Cf. A192872, A192232, A192744.
%K A192873 nonn,easy
%O A192873 0,3
%A A192873 _Clark Kimberling_, Jul 11 2011
%E A192873 More terms from _Colin Barker_, Apr 01 2014