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
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