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A192974 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
0, 1, 4, 14, 37, 84, 172, 329, 600, 1058, 1821, 3080, 5144, 8513, 13996, 22902, 37349, 60764, 98692, 160105, 259520, 420426, 680829, 1102224, 1784112, 2887489, 4672852, 7561694, 12236005, 19799268, 32036956, 51838025, 83876904, 135716978 (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+3*x^2)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014

a(n) = Fibonacci(n+7) + Lucas(n+3) - 2*n*(n+4) - 17. - Ehren Metcalfe, Jul 14 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}] (* A192973 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192974 *)

(* Additional programs *)

Table[Fibonacci[n+7] +LucasL[n+3] -2n(n+4) -17, {n, 0, 40}] (* Vincenzo Librandi, Jul 15 2019 *)

PROG

(PARI) a(n)=fibonacci(n+7) + fibonacci(2*n+6)/fibonacci(n+3) - 2*n*(n+4) - 17 \\ Richard N. Smith, Jul 14 2019

(MAGMA) [Fibonacci(n+7)+Lucas(n+3)-2*n*(n+4)-17: n in [0..40]]; // Vincenzo Librandi, Jul 15 2019

(Sage) f=fibonacci; [f(n+6)+3*f(n+4) -(2*n^2+8*n+17) for n in (0..40)] # G. C. Greubel, Jul 24 2019

(GAP) F:=Fibonacci;; List([0..40], n-> F(n+6)+3*F(n+4) -(2*n^2+8*n+17)); # G. C. Greubel, Jul 24 2019

CROSSREFS

Cf. A000032, A000045, A192232, A192744, A192951, A192971.

Sequence in context: A027166 A126943 A209399 * A187428 A316878 A036368

Adjacent sequences:  A192971 A192972 A192973 * A192975 A192976 A192977

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified June 23 10:45 EDT 2021. Contains 345397 sequences. (Running on oeis4.)