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A192808 Constant term in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2. See Comments.

%I A192808 #22 Jun 12 2025 22:35:13
%S A192808 1,2,6,26,126,618,3022,14746,71902,350538,1708910,8331130,40615294,
%T A192808 198004778,965298958,4705957722,22942154782,111845982474,545263681710,
%U A192808 2658231220538,12959222223038,63177890368490,308000415667278,1501542003033370
%N A192808 Constant term in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2.  See Comments.
%C A192808 For discussions of polynomial reduction, see A192232 and A192744.
%C A192808 If the reduction (x^2 + c)^n by x^3 -> x^2 + c is applied to the polynomials (x^2+c)^n for c=1 instead of c=2, the results are as follows:
%C A192808 A052554: constant terms,
%C A192808 A052529: coefficients of x,
%C A192808 A124820: coefficients of x^2.
%C A192808 Those three sequences satisfy the recurrence:
%C A192808 u(n) = 4*u(n-1) - 3*u(n-2) + u(n-3).
%H A192808 G. C. Greubel, <a href="/A192808/b192808.txt">Table of n, a(n) for n = 0..1000</a>
%H A192808 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-12,8).
%F A192808 a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).
%F A192808 G.f.: (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3). - _Colin Barker_, Jul 26 2012
%t A192808 q = x^3; s = x^2 + 2; z = 40;
%t A192808 p[n_, x_] := (x^2 + 2)^n;
%t A192808 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192808 reduce[{p1_, q_, s_, x_}] :=
%t A192808 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192808        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192808 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192808 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192808 *)
%t A192808 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192809 *)
%t A192808 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192810 *)
%t A192808 uu = u2/2  (* A192811 *)
%t A192808 LinearRecurrence[{7,-12,8}, {1,2,6}, 50] (* _G. C. Greubel_, Jan 02 2019 *)
%o A192808 (PARI) my(x='x+O('x^30)); Vec((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)) \\ _G. C. Greubel_, Jan 02 2019
%o A192808 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3) )); // _G. C. Greubel_, Jan 02 2019
%o A192808 (Sage) ((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 02 2019
%o A192808 (GAP) a:=[1,2,6];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # _Muniru A Asiru_, Jan 02 2019
%Y A192808 Cf. A192744, A192232, A192808 - A192811.
%K A192808 nonn,easy
%O A192808 0,2
%A A192808 _Clark Kimberling_, Jul 10 2011