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
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