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Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
3

%I #14 Sep 08 2022 08:45:58

%S 1,1,5,11,23,43,77,133,225,375,619,1015,1657,2697,4381,7107,11519,

%T 18659,30213,48909,79161,128111,207315,335471,542833,878353,1421237,

%U 2299643,3720935,6020635,9741629,15762325,25504017,41266407,66770491

%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^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="/A192954/b192954.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 From _R. J. Mathar_, May 08 2014: (Start)

%F G.f.: (1 -2*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2).

%F a(n) - a(n-1) = A168674(n-1). (End)

%F a(n) = 2*Lucas(n+2) - (2*n+5). - _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 *)

%t Table[2*LucasL[n+2]-(2*n+5), {n,0,40}] (* _G. C. Greubel_, Jul 12 2019 *)

%t LinearRecurrence[{3,-2,-1,1},{1,1,5,11},40] (* _Harvey P. Dale_, Jan 13 2022 *)

%o (PARI) vector(40, n, n--; f=fibonacci; 2*(f(n+3)+f(n+1))-(2*n+5)) \\ _G. C. Greubel_, Jul 12 2019

%o (Magma) [2*Lucas(n+2)-(2*n+5): n in [0..40]]; // _G. C. Greubel_, Jul 12 2019

%o (Sage) [2*lucas_number2(n+2,1,-1)-(2*n+5) for n in (0..40)] # _G. C. Greubel_, Jul 12 2019

%o (GAP) List([0..40], n-> 2*Lucas(1,-1,n+2)[2]-(2*n+5)); # _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