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A192234 a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,1,0,1.

%I A192234 #26 Sep 08 2022 08:45:57
%S A192234 0,1,0,1,4,9,28,81,232,673,1944,5617,16236,46921,135604,391905,
%T A192234 1132624,3273345,9460144,27340321,79014996,228357577,659965644,
%U A192234 1907336113,5512303672,15930853281,46041020488,133061018769,384553481404,1111380188041
%N A192234 a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,1,0,1.
%C A192234 With a different offset, constant term of the reduction of the n-th 1st-kind Chebyshev polynomial by x^2->x+1. See A192232.
%H A192234 Colin Barker, <a href="/A192234/b192234.txt">Table of n, a(n) for n = 0..1000</a>
%H A192234 H. S. M. Coxeter, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002022109">Loxodromic sequences of tangent spheres</a>, Aequationes Mathematicae, 1.1-2 (1968): 104-121. See p. 112.
%H A192234 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,-1).
%F A192234 G.f.: x*(1 - 2*x - x^2) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - _Colin Barker_, Feb 09 2012 and Sep 09 2018
%t A192234 q[x_]:= x + 1;
%t A192234 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192234 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, ChebyshevT[n, x]]]], {n, 1, 40}];
%t A192234 Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]  (* A192234 *)
%t A192234 Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A071101 *)
%t A192234 (* _Peter J. C. Moses_, Jun 25 2011 *)
%o A192234 (PARI) a(n)=my(t=polchebyshev(n));while(poldegree(t)>1, t=substpol(t, x^2,x+1));subst(t,x,0) \\ _Charles R Greathouse IV_, Feb 09 2012
%o A192234 (PARI) concat(0, Vec(x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4) + O(x^40))) \\ _Colin Barker_, Sep 09 2018
%o A192234 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4) )); // _G. C. Greubel_, Jul 29 2019
%o A192234 (Sage) (x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 29 2019
%o A192234 (GAP) a:=[0,1,0,1];; for n in [5..40] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3] -a[n-4]; od; a; # _G. C. Greubel_, Jul 29 2019
%Y A192234 Cf. A192232.
%K A192234 nonn,easy
%O A192234 0,5
%A A192234 _Clark Kimberling_, Jun 26 2011
%E A192234 Entry revised (with new offset and initial terms) by _N. J. A. Sloane_, Sep 03 2018