A192236 Coefficient of x in the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

2, 4, 12, 36, 102, 296, 856, 2472, 7146, 20652, 59684, 172492, 498510, 1440720, 4163760, 12033488, 34777426, 100508628, 290475324, 839489268, 2426169014, 7011758584, 20264358408, 58565082744, 169256230458, 489159584636, 1413697437268

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, A192235, A192237.

Programs

GAP

a:=[2,4,12,36];; 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:=[2,4,12,36]; [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; m:=40;
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, m}];
Table[Coefficient[Part[t, n], x, 0], {n, m}] (* A192235 *)
Table[Coefficient[Part[t, n], x, 1], {n, m}] (* A192236 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, m}] (* A192237 *)
(* Peter J. C. Moses, Jun 25 2011 *)
LinearRecurrence[{2,2,2,-1}, {2,4,12,36}, 40] (* G. C. Greubel, Jul 30 2019 *)

PARI

m=40; v=concat([2,4,12,36], 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

def a(n):
    if (n==0): return 2
    elif (1 <= n <= 3): return 4*3^(n-1)
    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

a(n) = 2*A192237(n+2).

G.f.: 2*x/(1-2*x-2*x^2-2*x^3+x^4). - Colin Barker, Sep 12 2012