A192814 Constant term in the reduction of the polynomial (2*x+1)^n by x^3 -> x^2 + x + 1. See Comments.

1, 1, 1, 9, 49, 225, 1041, 4873, 22817, 106753, 499425, 2336585, 10931921, 51145825, 239289457, 1119533257, 5237818689, 24505519873, 114650876097, 536402551689, 2509598769265, 11741342323937, 54932733173713, 257006830281609

Offset

0,4

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

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

Index entries for linear recurrences with constant coefficients, signature (5,-3,7).

Formula

a(n) = 5*a(n-1) - 3*a(n-2) + 7*a(n-3).

G.f.: (1 -4*x -x^2) / (1 -5*x +3*x^2 -7*x^3). - R. J. Mathar, May 06 2014

Maple

seq(coeff(series((1-4*x-x^2)/(1-5*x+3*x^2-7*x^3), x, n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Jan 03 2019

Mathematica

q = x^3; s = x^2 + x + 1; z = 40;
p[n_, x_] := (2 x + 1)^n;
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}] (* A192814 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192815 *)
u2 = u2/2  (* A192816 *)
LinearRecurrence[{5, -3, 7}, {1, 1, 1}, 30] (* G. C. Greubel, Jan 03 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-4*x-x^2)/(1-5*x+3*x^2-7*x^3)) \\ G. C. Greubel, Jan 03 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-4*x-x^2)/(1-5*x+3*x^2-7*x^3) )); // G. C. Greubel, Jan 03 2019
(Sage) ((1-4*x-x^2)/(1-5*x+3*x^2-7*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 03 2019
(GAP) a:=[1, 1, 1];; for n in [4..25] do a[n]:=5*a[n-1]-3*a[n-2]+7*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 03 2019

Crossrefs

Cf. A192744, A192232, A192815.

Sequence in context: A012248 A080026 A060867 * A228018 A081655 A181539

Adjacent sequences: A192811 A192812 A192813 * A192815 A192816 A192817

Keyword

nonn

Author

Clark Kimberling, Jul 10 2011

Status

approved