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

Author

Clark Kimberling, Jun 26 2011

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A192232.

Programs

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

Magma

R:=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

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 *)

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

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

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

Extensions

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