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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
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.
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved