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A192236 Coefficient of x in the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

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

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 *)

PROG

(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

(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

(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

(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

CROSSREFS

Cf. A192232, A192235, A192237.

Sequence in context: A241208 A149837 A149838 * A149839 A149840 A224540

Adjacent sequences:  A192233 A192234 A192235 * A192237 A192238 A192239

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 26 2011

STATUS

approved

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Last modified January 22 07:30 EST 2020. Contains 331139 sequences. (Running on oeis4.)