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A192965
Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
3
0, 1, 1, 4, 11, 27, 58, 115, 215, 386, 673, 1149, 1932, 3213, 5301, 8696, 14207, 23143, 37622, 61071, 99035, 160486, 259941, 420889, 681336, 1102777, 1784713, 2888140, 4673555, 7562451, 12236818, 19800139, 32037887, 51839018, 83877961
OFFSET
0,4
COMMENTS
The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) - 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.
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 +5*x^2 -x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+4) + 3*Fibonacci(n+2) - (n^2 + 3*n + 6). - G. C. Greubel, Jul 11 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(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}] (* A192964 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192965 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n+4]+3*F[n+2] -(n^2+3*n+6), {n, 0, 40}]] (* G. C. Greubel, Jul 11 2019 *)
PROG
(PARI) vector(40, n, n--; f=fibonacci; f(n+4)+3*f(n+2)-(n^2+3*n+6)) \\ G. C. Greubel, Jul 11 2019
(Magma) F:=Fibonacci; [F(n+4) +3*F(n+2) -(n^2+3*n+6): n in [0..40]]; // G. C. Greubel, Jul 11 2019
(Sage) f=fibonacci; [f(n+4) +3*f(n+2) -(n^2+3*n+6) for n in (0..40)] # G. C. Greubel, Jul 11 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+4) +3*F(n+2) -(n^2+3*n+6)); # G. C. Greubel, Jul 11 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved