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%I #42 Sep 08 2022 08:45:57
%S 0,1,3,9,22,48,96,181,327,573,982,1656,2760,4561,7491,12249,19966,
%T 32472,52728,85525,138615,224541,363598,588624,952752,1541953,2495331,
%U 4037961,6534022,10572768,17107632,27681301,44789895,72472221,117263206
%N Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) +1 +n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.
%H Vincenzo Librandi, <a href="/A192389/b192389.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
%F G.f.: x*(1-x+2*x^2)/((1-x)^3*(1-x-x^2)). - _Colin Barker_, May 12 2014
%F a(n) = 3*Fibonacci(n+4) - n*(n+4) - 9. - _Ehren Metcalfe_, Jul 13 2019
%t (* First program *)
%t p[0, x]:= 1; p[n_, x_]:= x*p[n-1, x] +n^2 +1;
%t Table[Expand[p[n, x]], {n, 0, 7}]
%t reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192953 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192389 *)
%t (* Additional programs *)
%t CoefficientList[Series[x*(1-x+2*x^2)/((1-x)^3*(1-x-x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, May 13 2014 *)
%t Table[3*Fibonacci[n+4] -n*(n+4)-9, {n,0,40}] (* _G. C. Greubel_, Jul 24 2019 *)
%o (PARI) Vec(x*(1-x+2*x^2)/((1-x)^3*(1-x-x^2)) + O(x^40)) \\ _Colin Barker_, May 12 2014
%o (PARI) vector(40, n, n--; 3*fibonacci(n+2)-n*(n+4)-9) \\ _G. C. Greubel_, Jul 24 2019
%o (Magma) [3*Fibonacci(n+2)-n*(n+4)-9: n in [0..40]]; // _G. C. Greubel_, Jul 24 2019
%o (Sage) [3*fibonacci(n+2)-n*(n+4)-9 for n in (0..40)] # _G. C. Greubel_, Jul 24 2019
%o (GAP) List([0..40], n-> 3*Fibonacci(n+2)-n*(n+4)-9); # _G. C. Greubel_, Jul 24 2019
%Y Cf. A000045, A192232, A192744, A192951.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Jul 13 2011
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