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A192976
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Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
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3
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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MATHEMATICA
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(* 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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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