A192972 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 3, 12, 33, 77, 160, 309, 567, 1004, 1733, 2937, 4912, 8137, 13387, 21916, 35753, 58181, 94512, 153341, 248575, 402716, 652173, 1055857, 1709088, 2766097, 4476435, 7243884, 11721777, 18967229, 30690688, 49659717, 80352327, 130014092

Offset

0,3

Comments

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

a(n) = 4*Fibonacci(n+4) + Lucas(n+3) - 2*(n^2+4*n+8). - G. C. Greubel, Jul 24 2019

Mathematica

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + 2*n^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}] (* A192971 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192972 *)
(* Additional programs *)
With[{F = Fibonacci}, Table[5*F[n+4]+F[n+2] -2*(n^2+4*n+8), {n, 0, 40}]] (* G. C. Greubel, Jul 24 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; 5*f(n+4)+f(n+2) -2*(n^2+4*n+8)) \\ G. C. Greubel, Jul 24 2019
(Magma) F:=Fibonacci; [5*F(n+4)+F(n+2) -2*(n^2+4*n+8): n in [0..40]]; // G. C. Greubel, Jul 24 2019
(Sage) f=fibonacci; [5*f(n+4)+f(n+2) -2*(n^2+4*n+8) for n in (0..40)] # G. C. Greubel, Jul 24 2019
(GAP) F:=Fibonacci;; List([0..40], n-> 5*F(n+4)+F(n+2) -2*(n^2+4*n+8)); # G. C. Greubel, Jul 24 2019

Crossrefs

Cf. A000032, A000045, A192232, A192744, A192951, A192971.

Sequence in context: A174963 A054602 A083725 * A159228 A054610 A183468

Adjacent sequences: A192969 A192970 A192971 * A192973 A192974 A192975

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved