|
%I #12 Sep 08 2022 08:45:58
%S 0,1,2,7,18,41,84,161,294,519,894,1513,2528,4185,6882,11263,18370,
%T 29889,48548,78761,127670,206831,334942,542257,877728,1420561,2298914,
%U 3720151,6019794,9740729,15761364,25502993,41265318,66769335,108035742
%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^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="/A192955/b192955.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 From _R. J. Mathar_, May 08 2014: (Start)
%F G.f.: x*(1 -2*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^3).
%F a(n) - a(n-1) = A192954(n-1). (End)
%F a(n) = 2*Lucas(n+3) - (n^2+4*n+8). - _G. C. Greubel_, Jul 12 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;
%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}] (* A192954 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192955 *)
%t (* Second program *)Table[2*LucasL[n+3]-(n^2+4*n+8), {n,0,40}] (* _G. C. Greubel_, Jul 12 2019 *)
%o (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
%o (Magma) [2*Lucas(n+3)-(n^2+4*n+8): n in [0..40]]; // _G. C. Greubel_, Jul 12 2019
%o (Sage) [2*lucas_number2(n+3,1,-1)-(n^2+4*n+8) for n in (0..40)] # _G. C. Greubel_, Jul 12 2019
%o (GAP) List([0..40], n-> 2*Lucas(1,-1,n+3)[2]-(n^2+4*n+8)); # _G. C. Greubel_, Jul 12 2019
%Y Cf. A000032, A000045, A192232, A192744, A192951, A192955.
%K nonn
%O 0,3
%A _Clark Kimberling_, Jul 13 2011
|
