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A192980 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
0, 1, 2, 6, 15, 34, 70, 135, 248, 440, 761, 1292, 2164, 3589, 5910, 9682, 15803, 25726, 41802, 67835, 109980, 178196, 288597, 467256, 756360, 1224169, 1981130, 3205950, 5187783, 8394490, 13583086, 21978447, 35562464, 57541904, 93105425 (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.

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

a(n) = Fibonacci(n+4) + Lucas(n+3) - (n^2 + 3*n + 7). - 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] +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}] (* A192979 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192980 *)

(* Additional programs *)

CoefficientList[Series[x*(1-2*x+3*x^2)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)

Table[Fibonacci[n+4]+LucasL[n+3] -(n^2+3*n+7), {n, 0, 40}] (* G. C. Greubel, Jul 24 2019 *)

PROG

(PARI) vector(40, n, n--; f=fibonacci; 2*f(n+4)+f(n+2) -(n^2+3*n+7)) \\ G. C. Greubel, Jul 24 2019

(MAGMA) [Fibonacci(n+4)+Lucas(n+3)-(n^2+3*n+7): n in [0..40]]; // G. C. Greubel, Jul 24 2019

(Sage) f=fibonacci; [2*f(n+4)+f(n+2) -(n^2+3*n+7) for n in (0..40)] # G. C. Greubel, Jul 24 2019

(GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+4)+F(n+2) -(n^2+3*n+7)); # G. C. Greubel, Jul 24 2019

CROSSREFS

Cf. A000032, A000045, A192232, A192744, A192951, A192979.

Sequence in context: A262151 A246320 A101352 * A281970 A221719 A095380

Adjacent sequences:  A192977 A192978 A192979 * A192981 A192982 A192983

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified July 5 16:07 EDT 2020. Contains 335473 sequences. (Running on oeis4.)