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A192978 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
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 (list; graph; refs; listen; history; text; internal format)
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 *)

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

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Last modified October 23 00:24 EDT 2020. Contains 337962 sequences. (Running on oeis4.)