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%I #29 Sep 08 2022 08:45:57
%S 0,0,0,1,2,6,18,51,148,428,1236,3573,10326,29842,86246,249255,720360,
%T 2081880,6016744,17388713,50254314,145237662,419744634,1213084507,
%U 3505879292,10132179204,29282541372,84628115229,244579792318,706848718634,2042830710990,5903890328655,17062559724240,49311712809136,142513495013072
%N a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,0,0,1.
%H Colin Barker, <a href="/A192237/b192237.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 (2,2,2,-1).
%F G.f.: x^3 / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - _Colin Barker_, Sep 12 2012 and Sep 06 2018
%t q[x_]:= x + 1;
%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, ChebyshevU[n, x]]]], {n, 1, 40}];
%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}] (* A192235 *)
%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}] (* A192236 *)
%t Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}] (* A192237 *)
%t (* by _Peter J. C. Moses_, Jun 25 2011 *)
%t LinearRecurrence[{2,2,2,-1}, {0,0,0,1}, 40] (* _Vincenzo Librandi_, Sep 06 2018 *)
%o (PARI) concat(vector(3), Vec(x^3/(1-2*x-2*x^2-2*x^3+x^4) + O(x^40))) \\ _Colin Barker_, Sep 06 2018
%o (Magma) I:=[0,0,0,1]; [n le 4 select I[n] else 2*(Self(n-1)+Self(n-2) +Self(n-3))-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Sep 06 2018
%o (Sage) (x^3/(1-2*x-2*x^2-2*x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 30 2019
%o (GAP) a:=[0,0,0,1];; 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 30 2019
%Y Cf. A192232, A192235.
%Y With a different offset, equals (A192236)/2.
%Y Other sequences with this recurrence but different initial conditions: A192234, A317973, A317974, A317975, A317976.
%K nonn
%O 0,5
%A _Clark Kimberling_, Jun 26 2011
%E Entry revised (with new offset and initial terms) by _N. J. A. Sloane_, Sep 03 2018
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