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

Author

Clark Kimberling, Jul 10 2011

Keywords

• nonn

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).

Crossrefs

Cf. A192744, A192232, A192815.

Programs

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

Magma

m:=30; R:=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

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 *)

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

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

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