A192234 a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,1,0,1.

0, 1, 0, 1, 4, 9, 28, 81, 232, 673, 1944, 5617, 16236, 46921, 135604, 391905, 1132624, 3273345, 9460144, 27340321, 79014996, 228357577, 659965644, 1907336113, 5512303672, 15930853281, 46041020488, 133061018769, 384553481404, 1111380188041

Offset

0,5

Comments

With a different offset, constant term of the reduction of the n-th 1st-kind Chebyshev polynomial by x^2->x+1. See A192232.

Links

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

H. S. M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae, 1.1-2 (1968): 104-121. See p. 112.

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

Formula

G.f.: x*(1 - 2*x - x^2) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Feb 09 2012 and Sep 09 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] &, ChebyshevT[n, x]]]], {n, 1, 40}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]  (* A192234 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A071101 *)
(* Peter J. C. Moses, Jun 25 2011 *)

Prog

(PARI) a(n)=my(t=polchebyshev(n)); while(poldegree(t)>1, t=substpol(t, x^2, x+1)); subst(t, x, 0) \\ Charles R Greathouse IV, Feb 09 2012
(PARI) concat(0, Vec(x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4) + O(x^40))) \\ Colin Barker, Sep 09 2018
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4) )); // G. C. Greubel, Jul 29 2019
(Sage) (x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019
(GAP) a:=[0, 1, 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 29 2019

Crossrefs

Cf. A192232.

Sequence in context: A000368 A232765 A094255 * A069563 A352878 A276984

Adjacent sequences: A192231 A192232 A192233 * A192235 A192236 A192237

Keyword

nonn,easy

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