A192964 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

1, 0, 3, 7, 16, 31, 57, 100, 171, 287, 476, 783, 1281, 2088, 3395, 5511, 8936, 14479, 23449, 37964, 61451, 99455, 160948, 260447, 421441, 681936, 1103427, 1785415, 2888896, 4674367, 7563321, 12237748, 19801131, 32038943, 51840140, 83879151

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) - n + n^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 +5*x^2 -x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014

a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - 2*(n+2). - G. C. Greubel, Jul 11 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(n-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}] (* A192964 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192965 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n+3]+3*F[n+1] -2*(n+2), {n, 0, 40}]] (* G. C. Greubel, Jul 11 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; f(n+3)+3*f(n+1)-2*(n+2)) \\ G. C. Greubel, Jul 11 2019
(Magma) F:=Fibonacci; [F(n+3) +3*F(n+1) -2*(n+2): n in [0..40]]; // G. C. Greubel, Jul 11 2019
(Sage) f=fibonacci; [f(n+3) +3*f(n+1) -2*(n+2) for n in (0..40)] # G. C. Greubel, Jul 11 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+3) +3*F(n+1) -2*(n+2)); # G. C. Greubel, Jul 11 2019

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192965.

Sequence in context: A240739 A301117 A000412 * A293351 A179904 A298311

Adjacent sequences: A192961 A192962 A192963 * A192965 A192966 A192967

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved