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A192875 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

%I A192875 #17 Sep 08 2022 08:45:58
%S A192875 0,1,3,11,37,119,391,1257,4087,13195,42757,138271,447615,1448249,
%T A192875 4687071,15166963,49082501,158832391,513995543,1663319433,5382623015,
%U A192875 17418520571,56367538373,182409150671,590288468367,1910213517529
%N A192875 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.
%C A192875 The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1.  See A192872.
%H A192875 G. C. Greubel, <a href="/A192875/b192875.txt">Table of n, a(n) for n = 0..1000</a>
%H A192875 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,6,-5,-6,4).
%F A192875 a(n) = 2*a(n-1) + 6*a(n-2) - 5*a(n-3) - 6*a(n-4) + 4*a(n-5).
%F A192875 G.f.: x*(1+2*x)*(1-x+x^2) / ( (1-x)*(1+x-x^2)*(1-2*x-4*x^2)). - _R. J. Mathar_, May 06 2014
%p A192875 seq(coeff(series(x*(1+2*x)*(1-x+x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Jan 08 2019
%t A192875 (See A192874.)
%t A192875 LinearRecurrence[{2,6,-5,-6,4}, {0,1,3,11,37}, 30] (* _G. C. Greubel_, Jan 08 2019 *)
%o A192875 (PARI) my(x='x+O('x^30)); concat([0], Vec(x*(1+2*x)*(1-x+x^2)/((1-x)*(1+ x-x^2)*(1-2*x-4*x^2)))) \\ _G. C. Greubel_, Jan 08 2019
%o A192875 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+2*x)*(1-x+x^2)/((1-x)*(1+ x-x^2)*(1-2*x-4*x^2)) )); // _G. C. Greubel_, Jan 08 2019
%o A192875 (Sage) (x*(1+2*x)*(1-x+x^2)/((1-x)*(1+ x-x^2)*(1-2*x-4*x^2))).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 08 2019
%o A192875 (GAP) a:=[0,1,3,11,37];; for n in [6..30] do a[n]:=2*a[n-1]+6*a[n-2] - 5*a[n-3]-6*a[n-4]+4*a[n-5]; od; a; # _G. C. Greubel_, Jan 08 2019
%Y A192875 Cf. A192872, A192874.
%K A192875 nonn
%O A192875 0,3
%A A192875 _Clark Kimberling_, Jul 11 2011