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%I #19 Sep 08 2022 08:45:58
%S 1,1,4,9,19,36,65,113,192,321,531,872,1425,2321,3772,6121,9923,16076,
%T 26033,42145,68216,110401,178659,289104,467809,756961,1224820,1981833,
%U 3206707,5188596,8395361,13584017,21979440,35563521,57543027,93106616
%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) + 1 - 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="/A192979/b192979.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.: (1-2*x+3*x^2)/((1-x)^2*(1-x-x^2)). - _Colin Barker_, May 11 2014
%F a(n) = Fibonacci(n+3) + Lucas(n+2) - 2*(n+2). - _G. C. Greubel_, Jul 24 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^2-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}] (* A192979 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192980 *)
%t (* Additional programs *)
%t Table[Fibonacci[n+3]+LucasL[n+2] -2*(n+2), {n,0,40}] (* _G. C. Greubel_, Jul 24 2019 *)
%o (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+3)+f(n+1) -2*(n+2)) \\ _G. C. Greubel_, Jul 24 2019
%o (Magma) [Fibonacci(n+3)+Lucas(n+2)-2*(n+2): n in [0..40]]; // _G. C. Greubel_, Jul 24 2019
%o (Sage) f=fibonacci; [2*f(n+3)+f(n+1) -2*(n+2) for n in (0..40)] # _G. C. Greubel_, Jul 24 2019
%o (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+3)+F(n+1) -2*(n+2)); # _G. C. Greubel_, Jul 24 2019
%Y Cf. A000032, A000045, A192232, A192744, A192951, A192980.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Jul 13 2011
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