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A192976 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
0, 1, 2, 10, 29, 70, 148, 289, 534, 950, 1645, 2794, 4680, 7761, 12778, 20930, 34157, 55598, 90332, 146577, 237630, 385006, 623517, 1009490, 1634064, 2644705, 4280018, 6926074, 11207549, 18135190, 29344420, 47481409, 76827750, 124311206 (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 + 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*(1-2*x+7*x^2-2*x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014

a(n) = Fibonacci(n+4) + 3*Lucas(n+3) - (2*n^2 + 8*n + 15). - 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 -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}] (* A192975 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192976 *)

(* Additional programs *)

Table[Fibonacci[n+4]+3*LucasL[n+3] -(2*n^2+8*n+15), {n, 0, 40}] (* G. C. Greubel, Jul 24 2019 *)

PROG

(PARI) vector(40, n, n--; f=fibonacci; 4*f(n+4)+3*f(n+2) -(2*n^2 + 8*n + 15)) \\ G. C. Greubel, Jul 24 2019

(MAGMA) [Fibonacci(n+4)+3*Lucas(n+3)-(2*n^2+8*n+15): n in [0..40]]; // G. C. Greubel, Jul 24 2019

(Sage) f=fibonacci; [4*f(n+4)+3*f(n+2) -(2*n^2+8*n+15) for n in (0..40)] # G. C. Greubel, Jul 24 2019

(GAP) F:=Fibonacci;; List([0..40], n-> 4*F(n+4)+3*F(n+2)-(2*n^2+8*n+15)); # G. C. Greubel, Jul 24 2019

CROSSREFS

Cf. A000032, A000045, A192232, A192744, A192951, A192975.

Sequence in context: A327696 A076438 A203551 * A264677 A101561 A279263

Adjacent sequences:  A192973 A192974 A192975 * A192977 A192978 A192979

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified July 5 17:04 EDT 2020. Contains 335473 sequences. (Running on oeis4.)