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A192955
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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, 7, 18, 41, 84, 161, 294, 519, 894, 1513, 2528, 4185, 6882, 11263, 18370, 29889, 48548, 78761, 127670, 206831, 334942, 542257, 877728, 1420561, 2298914, 3720151, 6019794, 9740729, 15761364, 25502993, 41265318, 66769335, 108035742
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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) + 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 +4*x^2 -x^3)/((1-x-x^2)*(1-x)^3).
a(n) - a(n-1) = A192954(n-1). (End)
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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] + n^2;
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}] (* A192954 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192955 *)
(* Second program *)Table[2*LucasL[n+3]-(n^2+4*n+8), {n, 0, 40}] (* G. C. Greubel, Jul 12 2019 *)
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PROG
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(PARI) vector(40, n, n--; f=fibonacci; 2*(f(n+4)+f(n+2))-(n^2+4*n+8)) \\ G. C. Greubel, Jul 12 2019
(Magma) [2*Lucas(n+3)-(n^2+4*n+8): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) [2*lucas_number2(n+3, 1, -1)-(n^2+4*n+8) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) List([0..40], n-> 2*Lucas(1, -1, n+3)[2]-(n^2+4*n+8)); # G. C. Greubel, Jul 12 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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