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A192235 Constant term of the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

%I A192235 #22 Sep 08 2022 08:45:57
%S A192235 0,3,8,21,64,183,528,1529,4416,12763,36888,106605,308096,890415,
%T A192235 2573344,7437105,21493632,62117747,179523624,518832901,1499454912,
%U A192235 4333505127,12524062256,36195211689,104606103232,302317249227,873713066040
%N A192235 Constant term of the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.
%C A192235 See A192232.
%H A192235 G. C. Greubel, <a href="/A192235/b192235.txt">Table of n, a(n) for n = 1..1000</a>
%H A192235 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,-1).
%F A192235 Empirical G.f.: x^2*(3-x)*(1+x)/(1-2*x-2*x^2-2*x^3+x^4). - _Colin Barker_, Sep 11 2012
%t A192235 q[x_]:= x + 1;
%t A192235 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192235 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, ChebyshevU[n, x]]]], {n, 1, 40}];
%t A192235 Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}] (* A192235 *)
%t A192235 Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}] (* A192236 *)
%t A192235 Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}] (* A192237 *)
%t A192235 (* by _Peter J. C. Moses_, Jun 25 2011 *)
%t A192235 LinearRecurrence[{2,2,2,-1}, {0,3,8,21}, 40] (* _G. C. Greubel_, Jul 30 2019 *)
%o A192235 (PARI) a(n)=my(t=polchebyshev(n,2));while(poldegree(t)>1, t=substpol(t, x^2,x+1));subst(t,x,0) \\ _Charles R Greathouse IV_, Feb 09 2012
%o A192235 (PARI) m=40; v=concat([0, 3, 8, 21], vector(m-4)); for(n=5, m, v[n] = 2*v[n-1]+2*v[n-2]+2*v[n-3]-v[n-4]); v \\ _G. C. Greubel_, Jul 30 2019
%o A192235 (Magma) I:=[0, 3, 8, 21]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) +2*Self(n-3) -Self(n-4): n in [1..40]]; // _G. C. Greubel_, Jul 30 2019
%o A192235 (Sage)
%o A192235 @cached_function
%o A192235 def a(n):
%o A192235     if (n==0): return 0
%o A192235     elif (1 <= n <= 3): return fibonacci(2*n+2)
%o A192235     else: return 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4)
%o A192235 [a(n) for n in (0..40)] # _G. C. Greubel_, Jul 30 2019
%o A192235 (GAP) a:=[0,3,8,21];; 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 30 2019
%Y A192235 Cf. A192232, A192236, A192237.
%K A192235 nonn
%O A192235 1,2
%A A192235 _Clark Kimberling_, Jun 26 2011