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A192981 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 5 17:04 EDT 2020. Contains 335473 sequences. (Running on oeis4.)