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A192237
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.
5
0, 0, 0, 1, 2, 6, 18, 51, 148, 428, 1236, 3573, 10326, 29842, 86246, 249255, 720360, 2081880, 6016744, 17388713, 50254314, 145237662, 419744634, 1213084507, 3505879292, 10132179204, 29282541372, 84628115229, 244579792318, 706848718634, 2042830710990, 5903890328655, 17062559724240, 49311712809136, 142513495013072
OFFSET
0,5
FORMULA
G.f.: x^3 / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Sep 12 2012 and Sep 06 2018
MATHEMATICA
q[x_]:= x + 1;
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, ChebyshevU[n, x]]]], {n, 1, 40}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}] (* A192235 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}] (* A192236 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}] (* A192237 *)
(* by Peter J. C. Moses, Jun 25 2011 *)
LinearRecurrence[{2, 2, 2, -1}, {0, 0, 0, 1}, 40] (* Vincenzo Librandi, Sep 06 2018 *)
PROG
(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
(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
(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
(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
CROSSREFS
With a different offset, equals (A192236)/2.
Other sequences with this recurrence but different initial conditions: A192234, A317973, A317974, A317975, A317976.
Sequence in context: A196593 A248735 A219136 * A034525 A018249 A374186
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 26 2011
EXTENSIONS
Entry revised (with new offset and initial terms) by N. J. A. Sloane, Sep 03 2018
STATUS
approved