A192922 - OEIS

%I #16 Sep 08 2022 08:45:58

%S 1,0,1,2,5,11,25,55,122,268,590,1295,2844,6240,13693,30039,65900,

%T 144559,317108,695595,1525829,3346965,7341695,16104238,35325142,

%U 77486710,169969295,372832346

%N Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

%C The titular polynomial is defined by p(n,x) = p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x.  For discussions of polynomial reduction, see A192232, A192744, and A192872.

%H G. C. Greubel, <a href="/A192922/b192922.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-3,-1).

%F a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - a(n-4).

%F G.f.: (1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4). - _R. J. Mathar_, May 08 2014

%t q = x^2; s = x + 1; z = 28;

%t p[0, x_] := 1; p[1, x_] := x;

%t p[n_, x_] := p[n - 1, x] + p[n - 2, x]*x^2;

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 +

%t        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}]

%t   (* A192922 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192923 *)

%t LinearRecurrence[{2,2,-3,-1}, {1,0,1,2}, 30] (* _G. C. Greubel_, Feb 06 2019 *)

%o (PARI) my(x='x+O(x^30)); Vec((1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4)) \\ _G. C. Greubel_, Feb 06 2019

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4) )); // _G. C. Greubel_, Feb 06 2019

%o (Sage) ((1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4)).series(x, 30).coefficients(x, sparse=False)

%o (GAP) a:=[1,0,1,2];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2] -3*a[n-3]-a[n-4]; od; a; # _G. C. Greubel_, Feb 06 2019

%Y Cf. A192232, A192744, A192923.

%K nonn

%O 0,4

%A _Clark Kimberling_, Jul 12 2011