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.

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

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

Cf. A192232, A192235.

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

Adjacent sequences: A192234 A192235 A192236 * A192238 A192239 A192240

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