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A192236 Coefficient of x in the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

%I A192236 #19 Sep 08 2022 08:45:57
%S A192236 2,4,12,36,102,296,856,2472,7146,20652,59684,172492,498510,1440720,
%T A192236 4163760,12033488,34777426,100508628,290475324,839489268,2426169014,
%U A192236 7011758584,20264358408,58565082744,169256230458,489159584636,1413697437268
%N A192236 Coefficient of x in the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.
%C A192236 See A192232.
%H A192236 G. C. Greubel, <a href="/A192236/b192236.txt">Table of n, a(n) for n = 1..1000</a>
%H A192236 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,-1).
%F A192236 a(n) = 2*A192237(n+2).
%F A192236 G.f.: 2*x/(1-2*x-2*x^2-2*x^3+x^4). - _Colin Barker_, Sep 12 2012
%t A192236 q[x_]:= x + 1; m:=40;
%t A192236 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192236 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, ChebyshevU[n, x]]]], {n, m}];
%t A192236 Table[Coefficient[Part[t, n], x, 0], {n, m}] (* A192235 *)
%t A192236 Table[Coefficient[Part[t, n], x, 1], {n, m}] (* A192236 *)
%t A192236 Table[Coefficient[Part[t, n]/2, x, 1], {n, m}] (* A192237 *)
%t A192236 (* _Peter J. C. Moses_, Jun 25 2011 *)
%t A192236 LinearRecurrence[{2,2,2,-1}, {2,4,12,36}, 40] (* _G. C. Greubel_, Jul 30 2019 *)
%o A192236 (PARI) m=40; v=concat([2,4,12,36], 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 A192236 (Magma) I:=[2,4,12,36]; [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 A192236 (Sage)
%o A192236 def a(n):
%o A192236     if (n==0): return 2
%o A192236     elif (1 <= n <= 3): return 4*3^(n-1)
%o A192236     else: return 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4)
%o A192236 [a(n) for n in (0..40)] # _G. C. Greubel_, Jul 30 2019
%o A192236 (GAP) a:=[2,4,12,36];; 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 A192236 Cf. A192232, A192235, A192237.
%K A192236 nonn
%O A192236 1,1
%A A192236 _Clark Kimberling_, Jun 26 2011