A192978 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 4, 12, 29, 62, 122, 227, 406, 706, 1203, 2020, 3356, 5533, 9072, 14816, 24129, 39218, 63654, 103215, 167250, 270886, 438599, 709992, 1149144, 1859737, 3009532, 4869972, 7880261, 12751046, 20632178, 33384155, 54017326, 87402538

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 1 + 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.

Define a triangle by T(n,0) = n*(n+1) + 1, T(n,n)=1, and T(r,c) = T(r-1,c-1) + T(r-2,c-1). The sum of the terms in row(n) is a(n+1). - J. M. Bergot, Apr 14 2013

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

a(n) = Lucas(n+5) - n*(n+5) - 11. - Ehren Metcalfe, Jul 13 2019

From Stefano Spezia, Jul 13 2019: (Start)

a(n) = (1/2)*(-22 + (11 - 5*sqrt(5))*((1/2)*(1 - sqrt(5)))^n + 11*((1/2)* (1 + sqrt(5)))^n + 5*sqrt(5)*((1/2)*(1 + sqrt(5)))^n - 10*n - 2*n^2).

E.g.f.: (1/2)*(2 + sqrt(5))*((-47 + 21*sqrt(5))*exp(-(1/2)*(-1 + sqrt(5))*x) + (3 + sqrt(5))*exp((1/2)*(1 + sqrt(5))*x) - 2*(-2 + sqrt(5))*exp(x)*(11 + 6*x + x^2)).

(End)

Mathematica

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] +n^2 +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}] (* A027181 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192978 *)
(* Additional programs *)
CoefficientList[Series[x*(1+x^2)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)
Table[LucasL[n+5] -(n^2+5*n+11), {n, 0, 40}] (* G. C. Greubel, Jul 24 2019 *)
LinearRecurrence[{4, -5, 1, 2, -1}, {0, 1, 4, 12, 29}, 40] (* Harvey P. Dale, Dec 24 2023 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; f(n+6)+f(n+4) -(n^2+5*n+11)) \\ G. C. Greubel, Jul 24 2019
(Magma) [Lucas(n+5)-(n^2+5*n+11): n in [0..40]]; // G. C. Greubel, Jul 24 2019
(Sage) [lucas_number2(n+5, 1, -1) -(n^2+5*n+11) for n in (0..40)] # G. C. Greubel, Jul 24 2019
(GAP) List([0..40], n-> Lucas(1, -1, n+5)[2] -(n^2+5*n+11)); # G. C. Greubel, Jul 24 2019

Crossrefs

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

Sequence in context: A174121 A128563 A227085 * A260546 A062421 A036889

Adjacent sequences: A192975 A192976 A192977 * A192979 A192980 A192981

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved