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%I #15 Sep 08 2022 08:45:58
%S 1,0,3,7,16,31,57,100,171,287,476,783,1281,2088,3395,5511,8936,14479,
%T 23449,37964,61451,99455,160948,260447,421441,681936,1103427,1785415,
%U 2888896,4674367,7563321,12237748,19801131,32038943,51840140,83879151
%N Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
%C 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.
%H G. C. Greubel, <a href="/A192964/b192964.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F G.f.: (1 -3*x +5*x^2 -x^3)/((1-x-x^2)*(1-x)^2). - _R. J. Mathar_, May 11 2014
%F a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - 2*(n+2). - _G. C. Greubel_, Jul 11 2019
%t (* First program *)
%t q = x^2; s = x + 1; z = 40;
%t p[0, x]:= 1;
%t p[n_, x_]:= x*p[n-1, x] + n(n-1);
%t Table[Expand[p[n, x]], {n, 0, 7}]
%t reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192964 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192965 *)
%t (* Second program *)
%t With[{F=Fibonacci}, Table[F[n+3]+3*F[n+1] -2*(n+2), {n,0,40}]] (* _G. C. Greubel_, Jul 11 2019 *)
%o (PARI) vector(40, n, n--; f=fibonacci; f(n+3)+3*f(n+1)-2*(n+2)) \\ _G. C. Greubel_, Jul 11 2019
%o (Magma) F:=Fibonacci; [F(n+3) +3*F(n+1) -2*(n+2): n in [0..40]]; // _G. C. Greubel_, Jul 11 2019
%o (Sage) f=fibonacci; [f(n+3) +3*f(n+1) -2*(n+2) for n in (0..40)] # _G. C. Greubel_, Jul 11 2019
%o (GAP) F:=Fibonacci;; List([0..40], n-> F(n+3) +3*F(n+1) -2*(n+2)); # _G. C. Greubel_, Jul 11 2019
%Y Cf. A000045, A192232, A192744, A192951, A192965.
%K nonn
%O 0,3
%A _Clark Kimberling_, Jul 13 2011
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