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A192874 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments. 3
1, 0, 4, 6, 26, 72, 246, 774, 2532, 8150, 26434, 85448, 276654, 895054, 2896788, 9373678, 30334682, 98163784, 317666758, 1027987894, 3326644036, 10765237670, 34837054674, 112735054856, 364818336766, 1180576879422 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1.  See A192872.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,6,-5,-6,4).

FORMULA

a(n) = 2*a(n-1) + 6*a(n-2) - 5*a(n-3) - 6*a(n-4) + 4*a(n-5).

G.f.: (x^2-x+1)*(4*x^2+x-1) / ( (x-1)*(x^2-x-1)*(4*x^2+2*x-1) ). - R. J. Mathar, May 06 2014

MATHEMATICA

q = x^2; s = x + 1; z = 26;

p[0, x_] := 1; p[1, x_] := x;

p[n_, x_] := p[n - 1, x]*x + 2*p[n - 2, x]*x^2 + 1;

Table[Expand[p[n, x]], {n, 0, 7}]

reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192874 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192875 *)

LinearRecurrence[{2, 6, -5, -6, 4}, {1, 0, 4, 6, 26}, 30] (* G. C. Greubel, Jan 08 2019 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((x^2-x+1)*(4*x^2+x-1)/((x-1)*(x^2-x-1)*( 4*x^2+2*x-1))) \\ G. C. Greubel, Jan 08 2019

(MAGMA) I:=[1, 0, 4, 6, 26]; [n le 5 select I[n] else 2*Self(n-1)+6*Self(n-2) -5*Self(n-3)-6*Self(n-4)+4*Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 08 2019

(Sage) ((x^2-x+1)*(4*x^2+x-1)/((x-1)*(x^2-x-1)*( 4*x^2+2*x-1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019

(GAP) a:=[1, 0, 4, 6, 26];; for n in [6..30] do a[n]:=2*a[n-1]+6*a[n-2] - 5*a[n-3]-6*a[n-4]+4*a[n-5]; od; a; # G. C. Greubel, Jan 08 2019

CROSSREFS

Cf. A192872, A192875.

Sequence in context: A028273 A024471 A075277 * A159557 A176756 A054094

Adjacent sequences:  A192871 A192872 A192873 * A192875 A192876 A192877

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 11 2011

STATUS

approved

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Last modified September 18 04:22 EDT 2021. Contains 347508 sequences. (Running on oeis4.)