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

Author

Clark Kimberling, Jun 26 2011

Keywords

• nonn

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).

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.

Programs

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

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

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 *)

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

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

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

Extensions

Entry revised (with new offset and initial terms) by N. J. A. Sloane, Sep 03 2018