A192981 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

1, 0, 2, 5, 12, 24, 45, 80, 138, 233, 388, 640, 1049, 1712, 2786, 4525, 7340, 11896, 19269, 31200, 50506, 81745, 132292, 214080, 346417, 560544, 907010, 1467605, 2374668, 3842328, 6217053, 10059440, 16276554, 26336057, 42612676, 68948800

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + (n-1)^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, 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 (3,-2,-1,1).

Formula

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

G.f.: (1 - 3*x + 4*x^2)/((1 - x)^2*(1 - x - x^2)). - Colin Barker, May 11 2014

a(n) = Lucas(n+2) + Fibonacci(n+1) - (2*n+3). - G. C. Greubel, Jul 25 2019

Mathematica

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + (n-1)^2;
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}] (* A192981 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192982 *)
(* Additional programs *)
Table[LucasL[n+2]_Fibonacci[n+1]-(2*n+3), {n, 0, 40}] (* G. C. Greubel, Jul 25 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; f(n+3)+2*f(n+1) -(2*n+3)) \\ G. C. Greubel, Jul 25 2019
(Magma) F:=Fibonacci; [F(n+3)+2*F(n+1) -(2*n+3): n in [0..40]]; // G. C. Greubel, Jul 25 2019
(Sage) f=fibonacci; [f(n+3)+2*f(n+1) -(2*n+3) for n in (0..40)] # G. C. Greubel, Jul 25 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+3)+2*F(n+1) -(2*n+3)); # G. C. Greubel, Jul 25 2019

Crossrefs

Cf. A000032, A000045, A192232, A192744, A192951, A192982.

Sequence in context: A116733 A116721 A294868 * A116713 A261369 A116714

Adjacent sequences: A192978 A192979 A192980 * A192982 A192983 A192984

Keyword

nonn,easy

Author

Clark Kimberling, Jul 14 2011

Status

approved