A192235 Constant term of the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

0, 3, 8, 21, 64, 183, 528, 1529, 4416, 12763, 36888, 106605, 308096, 890415, 2573344, 7437105, 21493632, 62117747, 179523624, 518832901, 1499454912, 4333505127, 12524062256, 36195211689, 104606103232, 302317249227, 873713066040

Offset: 1

Author

Clark Kimberling, Jun 26 2011

Keywords

• nonn

Comments

See A192232.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Crossrefs

Cf. A192232, A192236, A192237.

Programs

GAP

a:=[0,3,8,21];; 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

Magma

I:=[0, 3, 8, 21]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) +2*Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Jul 30 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] &, 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}, {0,3,8,21}, 40] (* G. C. Greubel, Jul 30 2019 *)

PARI

a(n)=my(t=polchebyshev(n,2));while(poldegree(t)>1, t=substpol(t, x^2,x+1));subst(t,x,0) \\ Charles R Greathouse IV, Feb 09 2012

PARI

m=40; v=concat([0, 3, 8, 21], vector(m-4)); for(n=5, m, v[n] = 2*v[n-1]+2*v[n-2]+2*v[n-3]-v[n-4]); v \\ G. C. Greubel, Jul 30 2019

Sage

@cached_function
def a(n):
    if (n==0): return 0
    elif (1 <= n <= 3): return fibonacci(2*n+2)
    else: return 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 30 2019

Formula

Empirical G.f.: x^2*(3-x)*(1+x)/(1-2*x-2*x^2-2*x^3+x^4). - Colin Barker, Sep 11 2012