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A192956
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Constant term of 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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1, 0, 4, 9, 20, 38, 69, 120, 204, 341, 564, 926, 1513, 2464, 4004, 6497, 10532, 17062, 27629, 44728, 72396, 117165, 189604, 306814, 496465, 803328, 1299844, 2103225, 3403124, 5506406, 8909589, 14416056, 23325708, 37741829, 61067604, 98809502
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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 + 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) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 -3*x +6*x^2 -2*x^3)/((1-x-x^2)*(1-x)^2).
a(n) -2*a(n+1) +a(n+2) = A022096(n-3). (End)
a(n) = Fibonacci(n+3) + 4*Fibonacci(n+1) - (2*n+5). - G. C. Greubel, Jul 12 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] + 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}] (* A192956 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192957 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n+3]+4*F[n+1]-(2*n+5), {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; f(n+3)+4*f(n+1)-(2*n+5)) \\ G. C. Greubel, Jul 12 2019
(Magma) F:=Fibonacci; [F(n+3)+4*F(n+1)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) f=fibonacci; [f(n+3)+4*f(n+1)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+3)+4*F(n+1)-(2*n+5)); # 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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