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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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 A276984 A210969

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

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Last modified January 24 04:35 EST 2020. Contains 331183 sequences. (Running on oeis4.)