A192982 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 1, 3, 8, 20, 44, 89, 169, 307, 540, 928, 1568, 2617, 4329, 7115, 11640, 18980, 30876, 50145, 81345, 131851, 213596, 345888, 559968, 906385, 1466929, 2373939, 3841544, 6216212, 10058540, 16275593, 26335033, 42611587, 68947644, 111560320

Offset

0,4

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

Vincenzo Librandi, 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+4*x^2)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 12 2014

a(n) = Lucas(n+3) + Fibonacci(n+2) - (n^2 + 2*n + 5). - 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 *)
CoefficientList[Series[x*(1-3*x+4*x^2)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)
Table[LucasL[n+3]+Fibonacci[n+2]-(n^2+2*n+5), {n, 0, 40}] (* G. C. Greubel, Jul 25 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2) -(n^2+2*n+5)) \\ G. C. Greubel, Jul 25 2019
(Magma) F:=Fibonacci; [F(n+4)+2*F(n+2) -(n^2+2*n+5): n in [0..40]]; // G. C. Greubel, Jul 25 2019
(Sage) f=fibonacci; [f(n+4)+2*f(n+2) -(n^2+2*n+5) for n in (0..40)] # G. C. Greubel, Jul 25 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+4)+2*F(n+2) -(n^2+2*n+5)); # G. C. Greubel, Jul 25 2019

Crossrefs

Cf. A000032, A000045, A192232, A192744, A192951, A192981.

Sequence in context: A027298 A000236 A109327 * A096585 A057765 A290866

Adjacent sequences: A192979 A192980 A192981 * A192983 A192984 A192985

Keyword

nonn,easy

Author

Clark Kimberling, Jul 14 2011

Status

approved