A192882 - OEIS

%I #20 Sep 08 2022 08:45:58

%S 0,1,3,14,51,205,792,3107,12117,47362,184965,722591,2822544,11025793,

%T 43069611,168242270,657200859,2567211037,10028243016,39173122739,

%U 153021167805,597743469778,2334953116653,9120979734623,35629097057568

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

%C The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = 2*x*p(n-1,x) + (x^2)*p(n-1,x).  See A192872.

%H G. C. Greubel, <a href="/A192882/b192882.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 (2,7,2,-1).

%F a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).

%F G.f.: x*(1+x+x^2) / ( 1-2*x-7*x^2-2*x^3+x^4 ). - _R. J. Mathar_, May 07 2014

%F a(n) = Fibonacci(n)*Pell-Lucas(n)/2, where Pell-Lucas(n) = A002203(n). - _G. C. Greubel_, Jul 29 2019

%t (* First program *)

%t q = x^2; s = x + 1; z = 25;

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

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

%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}] (* A192880 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192882 *)

%t FindLinearRecurrence[u1]

%t FindLinearRecurrence[u2]

%t (* Additional programs *)

%t LinearRecurrence[{2,7,2,-1}, {0,1,3,14}, 30] (* _G. C. Greubel_, Jan 08 2019 *)

%t Table[Fibonacci[n]*LucasL[n, 2]/2, {n,0,30}] (* _G. C. Greubel_, Jul 29 2019 *)

%o (PARI) my(x='x+O('x^30)); concat([0], Vec(x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3 +x^4))) \\ _G. C. Greubel_, Jan 08 2019

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3+x^4) )); // _G. C. Greubel_, Jan 08 2019

%o (Sage) (x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 08 2019

%o (GAP) a:=[0,1,3,14];; for n in [5..30] do a[n]:=2*a[n-1]+7*a[n-2] +2*a[n-3] -a[n-4]; od; a; # _G. C. Greubel_, Jan 08 2019

%Y Cf. A000045, A002203, A192872, A192880.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jul 11 2011