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A192966 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192966 #24 Sep 08 2022 08:45:58
%S A192966 0,1,2,6,14,30,59,110,197,343,585,983,1634,2695,4420,7220,11760,19116,
%T A192966 31029,50316,81535,132061,213827,346141,560244,906685,1467254,2374290,
%U A192966 3841922,6216618,10058975,16276058,26335529,42612115,68948205,111560915
%N A192966 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192966 The titular polynomials are defined recursively:  p(n,x) = x*p(n-1,x) + n(n+1)/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 A192966 Vincenzo Librandi, <a href="/A192966/b192966.txt">Table of n, a(n) for n = 0..400</a>
%H A192966 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A192966 a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
%F A192966 G.f.: x*(1 - 2*x + 3*x^2 - x^3)/((1-x-x^2)*(1-x)^3). - _R. J. Mathar_, May 11 2014
%F A192966 a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (n^2 + 5*n + 10)/2. - _G. C. Greubel_, Jul 11 2019
%t A192966 q = x^2; s = x + 1; z = 40;
%t A192966 p[0, x]:= 1;
%t A192966 p[n_, x_]:= x*p[n-1, x] + n(n+1)/2;
%t A192966 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192966 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192966 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192966 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A030119 *)
%t A192966 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192966 *)
%t A192966 LinearRecurrence[{4,-5,1,2,-1},{0,1,2,6,14},40] (* _Vincenzo Librandi_, Nov 16 2011 *)
%t A192966 Table[Fibonacci[n+4] +2*Fibonacci[n+2] -(n^2+5*n+10)/2, {n,0,40}] (* _G. C. Greubel_, Jul 11 2019 *)
%o A192966 (Magma) I:=[0, 1, 2, 6, 14]; [n le 5 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+2*Self(n-4)-Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Nov 16 2011
%o A192966 (Magma) F:=Fibonacci; [F(n+4) +2*F(n+2) -(n^2+5*n+10)/2: n in [0..40]]; // _G. C. Greubel_, Jul 11 2019
%o A192966 (PARI) vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(n^2+5*n+10)/2) \\ _G. C. Greubel_, Jul 11 2019
%o A192966 (Sage) f=fibonacci; [f(n+4) +2*f(n+2) -(n^2+5*n+10)/2 for n in (0..40)] # _G. C. Greubel_, Jul 11 2019
%o A192966 (GAP) F:=Fibonacci;; List([0..40], n-> F(n+4) +2*F(n+2) -(n^2+5*n+10)/2); # _G. C. Greubel_, Jul 11 2019
%Y A192966 Cf. A000045, A192232, A192744, A192951.
%K A192966 nonn,easy
%O A192966 0,3
%A A192966 _Clark Kimberling_, Jul 13 2011