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%I #25 Sep 08 2022 08:45:58
%S 0,1,3,9,21,44,85,156,276,476,806,1347,2230,3667,6001,9787,15923,
%T 25862,41955,68006,110170,178406,288828,467509,756636,1224469,1981455,
%U 3206301,5188161,8394896,13583521,21978912,35562960,57542432,93105986
%N Coefficient of x in 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+3)/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 Vincenzo Librandi, <a href="/A192970/b192970.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
%F G.f.: x*(1-x+2*x^2-x^3)/((1-x-x^2)*(1-x)^3). - _R. J. Mathar_, May 11 2014
%F a(n) = Fibonacci(n+4) + Lucas(n+3) - (n^2 + 7*n + 14)/2. - _Ehren Metcalfe_, Jul 13 2019
%t (* First progream *)
%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+3)/2;
%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}] (* A192969 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192970 *)
%t (* Additional programs *)
%t CoefficientList[Series[x*(1-x+2*x^2-x^3)/((1-x-x^2)*(1-x)^3), {x,0,40}], x] (* _Vincenzo Librandi_, Jul 13 2019 *)
%t Table[LucasL[n+3]+Fibonacci[n+4]-(n^2+7*n+14)/2, {n,0,40}] (* _G. C. Greubel_, Jul 24 2019 *)
%o (Magma) [Fibonacci(n+4)+Lucas(n+3)-(n^2+7*n+14)/2: n in [0..40]]; // _Vincenzo Librandi_, Jul 13 2019
%o (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+4)+f(n+2)-(n^2+7*n+14)/2) \\ _G. C. Greubel_, Jul 24 2019
%o (Sage) f=fibonacci; [2*f(n+4)+f(n+2)-(n^2+7*n+14)/2 for n in (0..40)] # _G. C. Greubel_, Jul 24 2019
%o (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+4)+F(n+2)-(n^2+7*n+14)/2); # _G. C. Greubel_, Jul 24 2019
%Y Cf. A000032, A000045, A192232, A192744, A192951, A192970.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Jul 13 2011
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