A192965 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 1, 4, 11, 27, 58, 115, 215, 386, 673, 1149, 1932, 3213, 5301, 8696, 14207, 23143, 37622, 61071, 99035, 160486, 259941, 420889, 681336, 1102777, 1784713, 2888140, 4673555, 7562451, 12236818, 19800139, 32037887, 51839018, 83877961

Offset

0,4

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 (4,-5,1,2,-1).

Formula

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).

G.f.: x*(1 -3*x +5*x^2 -x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014

a(n) = Fibonacci(n+4) + 3*Fibonacci(n+2) - (n^2 + 3*n + 6). - 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+4]+3*F[n+2] -(n^2+3*n+6), {n, 0, 40}]] (* G. C. Greubel, Jul 11 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; f(n+4)+3*f(n+2)-(n^2+3*n+6)) \\ G. C. Greubel, Jul 11 2019
(Magma) F:=Fibonacci; [F(n+4) +3*F(n+2) -(n^2+3*n+6): n in [0..40]]; // G. C. Greubel, Jul 11 2019
(Sage) f=fibonacci; [f(n+4) +3*f(n+2) -(n^2+3*n+6) for n in (0..40)] # G. C. Greubel, Jul 11 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+4) +3*F(n+2) -(n^2+3*n+6)); # G. C. Greubel, Jul 11 2019

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192964.

Sequence in context: A268775 A294840 A014630 * A305119 A035593 A295056

Adjacent sequences: A192962 A192963 A192964 * A192966 A192967 A192968

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved