A192963 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 3, 10, 25, 55, 110, 207, 373, 652, 1115, 1877, 3124, 5157, 8463, 13830, 22533, 36635, 59474, 96451, 156305, 253176, 409943, 663625, 1074120, 1738345, 2813115, 4552162, 7366033, 11919007, 19285910, 31205847, 50492749, 81699652

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

a(n) = 3*Fibonacci(n+3) + 4*Fibonacci(n+2) - (n^2 + 5*n +10). - G. C. Greubel, Jul 12 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}] (* A192962 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192963 *)
(* Second program *)
With[{F=Fibonacci}, Table[3*F[n+3]+4*F[n+2] -(n^2+5*n+10), {n, 0, 40}]] (* G. C. Greubel, Jul 11 2019 *)
LinearRecurrence[{4, -5, 1, 2, -1}, {0, 1, 3, 10, 25}, 50] (* Harvey P. Dale, Apr 03 2023 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; 3*f(n+4)+4*f(n+2)-(n^2+5*n+10)) \\ G. C. Greubel, Jul 12 2019
(Magma) F:=Fibonacci; [3*F(n+4) +4*F(n+2) -(n^2+5*n+10): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) f=fibonacci; [3*f(n+4) +4*f(n+2) -(n^2+5*n+10) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) F:=Fibonacci;; List([0..40], n-> 3*F(n+4) +4*F(n+2) -(n^2+5*n+10)); # G. C. Greubel, Jul 12 2019

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192962.

Sequence in context: A162607 A267574 A047667 * A000247 A097763 A034506

Adjacent sequences: A192960 A192961 A192962 * A192964 A192965 A192966

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved